Integrand size = 293, antiderivative size = 28 \[ \int \frac {-6 e^5-6 x+\left (-6 x-6 x^2-2 x^6+e^5 \left (-6 x-2 x^5\right )\right ) \log \left (\frac {2}{x}\right )+\left (6 e^5 x^4+6 x^5\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-6 e^5 x^3-6 x^4\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (2 e^5 x^2+2 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )}{\left (-e^5 x^4-x^5\right ) \log \left (\frac {2}{x}\right )+\left (3 e^5 x^3+3 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-3 e^5 x^2-3 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (e^5 x+x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )} \, dx=x^2-\frac {3}{\left (-x+\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^2} \]
Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {-6 e^5-6 x+\left (-6 x-6 x^2-2 x^6+e^5 \left (-6 x-2 x^5\right )\right ) \log \left (\frac {2}{x}\right )+\left (6 e^5 x^4+6 x^5\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-6 e^5 x^3-6 x^4\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (2 e^5 x^2+2 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )}{\left (-e^5 x^4-x^5\right ) \log \left (\frac {2}{x}\right )+\left (3 e^5 x^3+3 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-3 e^5 x^2-3 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (e^5 x+x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )} \, dx=x^2-\frac {3}{\left (-x+\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )\right )^2} \]
Integrate[(-6*E^5 - 6*x + (-6*x - 6*x^2 - 2*x^6 + E^5*(-6*x - 2*x^5))*Log[ 2/x] + (6*E^5*x^4 + 6*x^5)*Log[2/x]*Log[(5*Log[2/x])/(E^5 + x)] + (-6*E^5* x^3 - 6*x^4)*Log[2/x]*Log[(5*Log[2/x])/(E^5 + x)]^2 + (2*E^5*x^2 + 2*x^3)* Log[2/x]*Log[(5*Log[2/x])/(E^5 + x)]^3)/((-(E^5*x^4) - x^5)*Log[2/x] + (3* E^5*x^3 + 3*x^4)*Log[2/x]*Log[(5*Log[2/x])/(E^5 + x)] + (-3*E^5*x^2 - 3*x^ 3)*Log[2/x]*Log[(5*Log[2/x])/(E^5 + x)]^2 + (E^5*x + x^2)*Log[2/x]*Log[(5* Log[2/x])/(E^5 + 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.
| \(\displaystyle \int \frac {\left (6 x^5+6 e^5 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )+\left (-6 x^4-6 e^5 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )+\left (2 x^3+2 e^5 x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )+\left (-2 x^6+e^5 \left (-2 x^5-6 x\right )-6 x^2-6 x\right ) \log \left (\frac {2}{x}\right )-6 x-6 e^5}{\left (x^2+e^5 x\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )+\left (-x^5-e^5 x^4\right ) \log \left (\frac {2}{x}\right )+\left (3 x^4+3 e^5 x^3\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )+\left (-3 x^3-3 e^5 x^2\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x \log \left (\frac {2}{x}\right ) \left (x^5+e^5 \left (x^4+3\right )-3 \left (x+e^5\right ) x^3 \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )+3 \left (x+e^5\right ) x^2 \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )+3 x-\left (x+e^5\right ) x \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )+3\right )+6 \left (x+e^5\right )}{x \left (x+e^5\right ) \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 \left (x^2 \log \left (\frac {2}{x}\right )+x+\left (1+e^5\right ) x \log \left (\frac {2}{x}\right )+e^5\right )}{\left (x+e^5\right ) x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )\right )^3}+2 x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6 \int \frac {1}{\left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )\right )^3}dx+6 \left (1+e^5\right ) \int \frac {1}{\left (x+e^5\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )\right )^3}dx-6 e^5 \int \frac {1}{\left (x+e^5\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )\right )^3}dx+6 \int \frac {1}{x \log \left (\frac {2}{x}\right ) \left (x-\log \left (\frac {5 \log \left (\frac {2}{x}\right )}{x+e^5}\right )\right )^3}dx+x^2\) |
Int[(-6*E^5 - 6*x + (-6*x - 6*x^2 - 2*x^6 + E^5*(-6*x - 2*x^5))*Log[2/x] + (6*E^5*x^4 + 6*x^5)*Log[2/x]*Log[(5*Log[2/x])/(E^5 + x)] + (-6*E^5*x^3 - 6*x^4)*Log[2/x]*Log[(5*Log[2/x])/(E^5 + x)]^2 + (2*E^5*x^2 + 2*x^3)*Log[2/ x]*Log[(5*Log[2/x])/(E^5 + x)]^3)/((-(E^5*x^4) - x^5)*Log[2/x] + (3*E^5*x^ 3 + 3*x^4)*Log[2/x]*Log[(5*Log[2/x])/(E^5 + x)] + (-3*E^5*x^2 - 3*x^3)*Log [2/x]*Log[(5*Log[2/x])/(E^5 + x)]^2 + (E^5*x + x^2)*Log[2/x]*Log[(5*Log[2/ x])/(E^5 + x)]^3),x]
3.12.95.3.1 Defintions of rubi rules used
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. \(144\) vs. \(2(27)=54\).
Time = 45.92 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.18
| method | result | size |
| parallelrisch | \(-\frac {3-2 \,{\mathrm e}^{10} x \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )-x^{2} \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )^{2}+2 x^{3} \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )+x^{2} {\mathrm e}^{10}-x^{4}+{\mathrm e}^{10} \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )^{2}}{x^{2}-2 \ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right ) x +\ln \left (\frac {5 \ln \left (\frac {2}{x}\right )}{{\mathrm e}^{5}+x}\right )^{2}}\) | \(145\) |
| default | \(x^{2}-\frac {12}{x^{2} {\left (\frac {2 \ln \left (\frac {1}{x}\right )}{x}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1}\right ) \operatorname {csgn}\left (i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{{\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1}\right )}{x}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{{\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1}\right )^{2}}{x}+\frac {i \pi \,\operatorname {csgn}\left (i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{{\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1}\right )^{2}}{x}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{{\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{x \left ({\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1\right )}\right )}{x}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{x \left ({\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1\right )}\right )}^{2}}{x}-\frac {i \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{{\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1}\right )^{3}}{x}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{{\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1}\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{x \left ({\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1\right )}\right )}^{2}}{x}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{x \left ({\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1\right )}\right )}^{3}}{x}+\frac {2 \ln \left (5\right )}{x}+\frac {2 \ln \left (\ln \left (2\right )+\ln \left (\frac {1}{x}\right )\right )}{x}-\frac {2 \ln \left ({\mathrm e}^{5+\ln \left (\frac {1}{x}\right )}+1\right )}{x}-2\right )}^{2}}\) | \(424\) |
| parts | \(\text {Expression too large to display}\) | \(32585\) |
int(((2*x^2*exp(5)+2*x^3)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))^3+(-6*x^3*exp(5 )-6*x^4)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))^2+(6*x^4*exp(5)+6*x^5)*ln(2/x)*l n(5*ln(2/x)/(exp(5)+x))+((-2*x^5-6*x)*exp(5)-2*x^6-6*x^2-6*x)*ln(2/x)-6*ex p(5)-6*x)/((x*exp(5)+x^2)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))^3+(-3*x^2*exp(5 )-3*x^3)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))^2+(3*x^3*exp(5)+3*x^4)*ln(2/x)*l n(5*ln(2/x)/(exp(5)+x))+(-x^4*exp(5)-x^5)*ln(2/x)),x,method=_RETURNVERBOSE )
-(3-2*exp(5)^2*x*ln(5*ln(2/x)/(exp(5)+x))-x^2*ln(5*ln(2/x)/(exp(5)+x))^2+2 *x^3*ln(5*ln(2/x)/(exp(5)+x))+x^2*exp(5)^2-x^4+exp(5)^2*ln(5*ln(2/x)/(exp( 5)+x))^2)/(x^2-2*ln(5*ln(2/x)/(exp(5)+x))*x+ln(5*ln(2/x)/(exp(5)+x))^2)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {-6 e^5-6 x+\left (-6 x-6 x^2-2 x^6+e^5 \left (-6 x-2 x^5\right )\right ) \log \left (\frac {2}{x}\right )+\left (6 e^5 x^4+6 x^5\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-6 e^5 x^3-6 x^4\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (2 e^5 x^2+2 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )}{\left (-e^5 x^4-x^5\right ) \log \left (\frac {2}{x}\right )+\left (3 e^5 x^3+3 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-3 e^5 x^2-3 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (e^5 x+x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )} \, dx=\frac {x^{4} - 2 \, x^{3} \log \left (\frac {5 \, \log \left (\frac {2}{x}\right )}{x + e^{5}}\right ) + x^{2} \log \left (\frac {5 \, \log \left (\frac {2}{x}\right )}{x + e^{5}}\right )^{2} - 3}{x^{2} - 2 \, x \log \left (\frac {5 \, \log \left (\frac {2}{x}\right )}{x + e^{5}}\right ) + \log \left (\frac {5 \, \log \left (\frac {2}{x}\right )}{x + e^{5}}\right )^{2}} \]
integrate(((2*x^2*exp(5)+2*x^3)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^3+(-6* x^3*exp(5)-6*x^4)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^2+(6*x^4*exp(5)+6*x^ 5)*log(2/x)*log(5*log(2/x)/(exp(5)+x))+((-2*x^5-6*x)*exp(5)-2*x^6-6*x^2-6* x)*log(2/x)-6*exp(5)-6*x)/((x*exp(5)+x^2)*log(2/x)*log(5*log(2/x)/(exp(5)+ x))^3+(-3*x^2*exp(5)-3*x^3)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^2+(3*x^3*e xp(5)+3*x^4)*log(2/x)*log(5*log(2/x)/(exp(5)+x))+(-x^4*exp(5)-x^5)*log(2/x )),x, algorithm=\
(x^4 - 2*x^3*log(5*log(2/x)/(x + e^5)) + x^2*log(5*log(2/x)/(x + e^5))^2 - 3)/(x^2 - 2*x*log(5*log(2/x)/(x + e^5)) + log(5*log(2/x)/(x + e^5))^2)
Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {-6 e^5-6 x+\left (-6 x-6 x^2-2 x^6+e^5 \left (-6 x-2 x^5\right )\right ) \log \left (\frac {2}{x}\right )+\left (6 e^5 x^4+6 x^5\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-6 e^5 x^3-6 x^4\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (2 e^5 x^2+2 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )}{\left (-e^5 x^4-x^5\right ) \log \left (\frac {2}{x}\right )+\left (3 e^5 x^3+3 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-3 e^5 x^2-3 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (e^5 x+x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )} \, dx=x^{2} - \frac {3}{x^{2} - 2 x \log {\left (\frac {5 \log {\left (\frac {2}{x} \right )}}{x + e^{5}} \right )} + \log {\left (\frac {5 \log {\left (\frac {2}{x} \right )}}{x + e^{5}} \right )}^{2}} \]
integrate(((2*x**2*exp(5)+2*x**3)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))**3+(-6* x**3*exp(5)-6*x**4)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))**2+(6*x**4*exp(5)+6*x **5)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))+((-2*x**5-6*x)*exp(5)-2*x**6-6*x**2- 6*x)*ln(2/x)-6*exp(5)-6*x)/((x*exp(5)+x**2)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x ))**3+(-3*x**2*exp(5)-3*x**3)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))**2+(3*x**3* exp(5)+3*x**4)*ln(2/x)*ln(5*ln(2/x)/(exp(5)+x))+(-x**4*exp(5)-x**5)*ln(2/x )),x)
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 215, normalized size of antiderivative = 7.68 \[ \int \frac {-6 e^5-6 x+\left (-6 x-6 x^2-2 x^6+e^5 \left (-6 x-2 x^5\right )\right ) \log \left (\frac {2}{x}\right )+\left (6 e^5 x^4+6 x^5\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-6 e^5 x^3-6 x^4\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (2 e^5 x^2+2 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )}{\left (-e^5 x^4-x^5\right ) \log \left (\frac {2}{x}\right )+\left (3 e^5 x^3+3 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-3 e^5 x^2-3 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (e^5 x+x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )} \, dx=\frac {2 \, {\left (i \, \pi + \log \left (5\right )\right )} x^{3} - x^{4} - x^{2} \log \left (x + e^{5}\right )^{2} - x^{2} \log \left (-\log \left (2\right ) + \log \left (x\right )\right )^{2} + {\left (\pi ^{2} - 2 i \, \pi \log \left (5\right ) - \log \left (5\right )^{2}\right )} x^{2} + 2 \, {\left ({\left (i \, \pi + \log \left (5\right )\right )} x^{2} - x^{3}\right )} \log \left (x + e^{5}\right ) + 2 \, {\left ({\left (-i \, \pi - \log \left (5\right )\right )} x^{2} + x^{3} + x^{2} \log \left (x + e^{5}\right )\right )} \log \left (-\log \left (2\right ) + \log \left (x\right )\right ) + 3}{\pi ^{2} + 2 \, {\left (i \, \pi + \log \left (5\right )\right )} x - x^{2} - 2 i \, \pi \log \left (5\right ) - \log \left (5\right )^{2} + 2 \, {\left (i \, \pi - x + \log \left (5\right )\right )} \log \left (x + e^{5}\right ) - \log \left (x + e^{5}\right )^{2} + 2 \, {\left (-i \, \pi + x - \log \left (5\right ) + \log \left (x + e^{5}\right )\right )} \log \left (-\log \left (2\right ) + \log \left (x\right )\right ) - \log \left (-\log \left (2\right ) + \log \left (x\right )\right )^{2}} \]
integrate(((2*x^2*exp(5)+2*x^3)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^3+(-6* x^3*exp(5)-6*x^4)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^2+(6*x^4*exp(5)+6*x^ 5)*log(2/x)*log(5*log(2/x)/(exp(5)+x))+((-2*x^5-6*x)*exp(5)-2*x^6-6*x^2-6* x)*log(2/x)-6*exp(5)-6*x)/((x*exp(5)+x^2)*log(2/x)*log(5*log(2/x)/(exp(5)+ x))^3+(-3*x^2*exp(5)-3*x^3)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^2+(3*x^3*e xp(5)+3*x^4)*log(2/x)*log(5*log(2/x)/(exp(5)+x))+(-x^4*exp(5)-x^5)*log(2/x )),x, algorithm=\
(2*(I*pi + log(5))*x^3 - x^4 - x^2*log(x + e^5)^2 - x^2*log(-log(2) + log( x))^2 + (pi^2 - 2*I*pi*log(5) - log(5)^2)*x^2 + 2*((I*pi + log(5))*x^2 - x ^3)*log(x + e^5) + 2*((-I*pi - log(5))*x^2 + x^3 + x^2*log(x + e^5))*log(- log(2) + log(x)) + 3)/(pi^2 + 2*(I*pi + log(5))*x - x^2 - 2*I*pi*log(5) - log(5)^2 + 2*(I*pi - x + log(5))*log(x + e^5) - log(x + e^5)^2 + 2*(-I*pi + x - log(5) + log(x + e^5))*log(-log(2) + log(x)) - log(-log(2) + log(x)) ^2)
Leaf count of result is larger than twice the leaf count of optimal. 31752 vs. \(2 (27) = 54\).
Time = 16.30 (sec) , antiderivative size = 31752, normalized size of antiderivative = 1134.00 \[ \int \frac {-6 e^5-6 x+\left (-6 x-6 x^2-2 x^6+e^5 \left (-6 x-2 x^5\right )\right ) \log \left (\frac {2}{x}\right )+\left (6 e^5 x^4+6 x^5\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-6 e^5 x^3-6 x^4\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (2 e^5 x^2+2 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )}{\left (-e^5 x^4-x^5\right ) \log \left (\frac {2}{x}\right )+\left (3 e^5 x^3+3 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-3 e^5 x^2-3 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (e^5 x+x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )} \, dx=\text {Too large to display} \]
integrate(((2*x^2*exp(5)+2*x^3)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^3+(-6* x^3*exp(5)-6*x^4)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^2+(6*x^4*exp(5)+6*x^ 5)*log(2/x)*log(5*log(2/x)/(exp(5)+x))+((-2*x^5-6*x)*exp(5)-2*x^6-6*x^2-6* x)*log(2/x)-6*exp(5)-6*x)/((x*exp(5)+x^2)*log(2/x)*log(5*log(2/x)/(exp(5)+ x))^3+(-3*x^2*exp(5)-3*x^3)*log(2/x)*log(5*log(2/x)/(exp(5)+x))^2+(3*x^3*e xp(5)+3*x^4)*log(2/x)*log(5*log(2/x)/(exp(5)+x))+(-x^4*exp(5)-x^5)*log(2/x )),x, algorithm=\
(x^10*log(2)^2*log(2/x)^3 + 3*x^9*e^5*log(2)^2*log(2/x)^3 + 2*x^9*log(2)^2 *log(x + e^5)*log(2/x)^3 + 6*x^8*e^5*log(2)^2*log(x + e^5)*log(2/x)^3 + x^ 8*log(2)^2*log(x + e^5)^2*log(2/x)^3 + 3*x^7*e^5*log(2)^2*log(x + e^5)^2*l og(2/x)^3 - 2*x^10*log(2)*log(x)*log(2/x)^3 - 6*x^9*e^5*log(2)*log(x)*log( 2/x)^3 - 4*x^9*log(2)*log(x + e^5)*log(x)*log(2/x)^3 - 12*x^8*e^5*log(2)*l og(x + e^5)*log(x)*log(2/x)^3 - 2*x^8*log(2)*log(x + e^5)^2*log(x)*log(2/x )^3 - 6*x^7*e^5*log(2)*log(x + e^5)^2*log(x)*log(2/x)^3 + x^10*log(x)^2*lo g(2/x)^3 + 3*x^9*e^5*log(x)^2*log(2/x)^3 + 2*x^9*log(x + e^5)*log(x)^2*log (2/x)^3 + 6*x^8*e^5*log(x + e^5)*log(x)^2*log(2/x)^3 + x^8*log(x + e^5)^2* log(x)^2*log(2/x)^3 + 3*x^7*e^5*log(x + e^5)^2*log(x)^2*log(2/x)^3 - 2*x^9 *log(2)^2*log(5*log(2/x))*log(2/x)^3 - 6*x^8*e^5*log(2)^2*log(5*log(2/x))* log(2/x)^3 - 2*x^8*log(2)^2*log(x + e^5)*log(5*log(2/x))*log(2/x)^3 - 6*x^ 7*e^5*log(2)^2*log(x + e^5)*log(5*log(2/x))*log(2/x)^3 + 4*x^9*log(2)*log( x)*log(5*log(2/x))*log(2/x)^3 + 12*x^8*e^5*log(2)*log(x)*log(5*log(2/x))*l og(2/x)^3 + 4*x^8*log(2)*log(x + e^5)*log(x)*log(5*log(2/x))*log(2/x)^3 + 12*x^7*e^5*log(2)*log(x + e^5)*log(x)*log(5*log(2/x))*log(2/x)^3 - 2*x^9*l og(x)^2*log(5*log(2/x))*log(2/x)^3 - 6*x^8*e^5*log(x)^2*log(5*log(2/x))*lo g(2/x)^3 - 2*x^8*log(x + e^5)*log(x)^2*log(5*log(2/x))*log(2/x)^3 - 6*x^7* e^5*log(x + e^5)*log(x)^2*log(5*log(2/x))*log(2/x)^3 + x^8*log(2)^2*log(5* log(2/x))^2*log(2/x)^3 + 3*x^7*e^5*log(2)^2*log(5*log(2/x))^2*log(2/x)^...
Time = 14.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {-6 e^5-6 x+\left (-6 x-6 x^2-2 x^6+e^5 \left (-6 x-2 x^5\right )\right ) \log \left (\frac {2}{x}\right )+\left (6 e^5 x^4+6 x^5\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-6 e^5 x^3-6 x^4\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (2 e^5 x^2+2 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )}{\left (-e^5 x^4-x^5\right ) \log \left (\frac {2}{x}\right )+\left (3 e^5 x^3+3 x^4\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (-3 e^5 x^2-3 x^3\right ) \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )+\left (e^5 x+x^2\right ) \log \left (\frac {2}{x}\right ) \log ^3\left (\frac {5 \log \left (\frac {2}{x}\right )}{e^5+x}\right )} \, dx=x^2-\frac {3}{{\left (x-\ln \left (\frac {5\,\ln \left (\frac {2}{x}\right )}{x+{\mathrm {e}}^5}\right )\right )}^2} \]
int((6*x + 6*exp(5) + log(2/x)*(6*x + exp(5)*(6*x + 2*x^5) + 6*x^2 + 2*x^6 ) - log((5*log(2/x))/(x + exp(5)))*log(2/x)*(6*x^4*exp(5) + 6*x^5) - log(( 5*log(2/x))/(x + exp(5)))^3*log(2/x)*(2*x^2*exp(5) + 2*x^3) + log((5*log(2 /x))/(x + exp(5)))^2*log(2/x)*(6*x^3*exp(5) + 6*x^4))/(log(2/x)*(x^4*exp(5 ) + x^5) - log((5*log(2/x))/(x + exp(5)))*log(2/x)*(3*x^3*exp(5) + 3*x^4) + log((5*log(2/x))/(x + exp(5)))^2*log(2/x)*(3*x^2*exp(5) + 3*x^3) - log(( 5*log(2/x))/(x + exp(5)))^3*log(2/x)*(x*exp(5) + x^2)),x)