Integrand size = 262, antiderivative size = 38 \[ \int \frac {x (i \pi +\log (3))^4-2 (i \pi +\log (3))^4 \log (x)+\left (x (i \pi +\log (3))^4-(i \pi +\log (3))^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )+\left (-4 e^{4+x} x^3 (i \pi +\log (3))^2+8 e^{4+x} x^2 (i \pi +\log (3))^2 \log (x)\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{4+x} \left (-4 x^3+2 x^4\right ) (i \pi +\log (3))^2+e^{4+x} \left (4 x^2-2 x^3\right ) (i \pi +\log (3))^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (-8 e^{8+2 x} x^6+8 e^{8+2 x} x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}{\left (-4 x^6+4 x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx=\left (-e^{4+x}+\frac {(i \pi +\log (3))^2}{4 x^2 \log ^2\left (x-\log ^2(x)\right )}\right )^2 \] Output:
(1/4*(ln(3)+I*Pi)^2/x^2/ln(-ln(x)^2+x)^2-exp(4+x))^2
Time = 0.36 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \frac {x (i \pi +\log (3))^4-2 (i \pi +\log (3))^4 \log (x)+\left (x (i \pi +\log (3))^4-(i \pi +\log (3))^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )+\left (-4 e^{4+x} x^3 (i \pi +\log (3))^2+8 e^{4+x} x^2 (i \pi +\log (3))^2 \log (x)\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{4+x} \left (-4 x^3+2 x^4\right ) (i \pi +\log (3))^2+e^{4+x} \left (4 x^2-2 x^3\right ) (i \pi +\log (3))^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (-8 e^{8+2 x} x^6+8 e^{8+2 x} x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}{\left (-4 x^6+4 x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx=\frac {\left ((\pi -i \log (3))^2+4 e^{4+x} x^2 \log ^2\left (x-\log ^2(x)\right )\right )^2}{16 x^4 \log ^4\left (x-\log ^2(x)\right )} \] Input:
Integrate[(x*(I*Pi + Log[3])^4 - 2*(I*Pi + Log[3])^4*Log[x] + (x*(I*Pi + L og[3])^4 - (I*Pi + Log[3])^4*Log[x]^2)*Log[x - Log[x]^2] + (-4*E^(4 + x)*x ^3*(I*Pi + Log[3])^2 + 8*E^(4 + x)*x^2*(I*Pi + Log[3])^2*Log[x])*Log[x - L og[x]^2]^2 + (E^(4 + x)*(-4*x^3 + 2*x^4)*(I*Pi + Log[3])^2 + E^(4 + x)*(4* x^2 - 2*x^3)*(I*Pi + Log[3])^2*Log[x]^2)*Log[x - Log[x]^2]^3 + (-8*E^(8 + 2*x)*x^6 + 8*E^(8 + 2*x)*x^5*Log[x]^2)*Log[x - Log[x]^2]^5)/((-4*x^6 + 4*x ^5*Log[x]^2)*Log[x - Log[x]^2]^5),x]
Output:
((Pi - I*Log[3])^2 + 4*E^(4 + x)*x^2*Log[x - Log[x]^2]^2)^2/(16*x^4*Log[x - Log[x]^2]^4)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(38)=76\).
Time = 8.93 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {3041, 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.
| \(\displaystyle \int \frac {\left (8 e^{2 x+8} x^5 \log ^2(x)-8 e^{2 x+8} x^6\right ) \log ^5\left (x-\log ^2(x)\right )+\left (8 e^{x+4} x^2 (\log (3)+i \pi )^2 \log (x)-4 e^{x+4} x^3 (\log (3)+i \pi )^2\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{x+4} \left (2 x^4-4 x^3\right ) (\log (3)+i \pi )^2+e^{x+4} \left (4 x^2-2 x^3\right ) (\log (3)+i \pi )^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (x (\log (3)+i \pi )^4-(\log (3)+i \pi )^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )-2 (\log (3)+i \pi )^4 \log (x)+x (\log (3)+i \pi )^4}{\left (4 x^5 \log ^2(x)-4 x^6\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx\) |
| \(\Big \downarrow \) 3041 |
| \(\displaystyle \int \frac {\left (8 e^{2 x+8} x^5 \log ^2(x)-8 e^{2 x+8} x^6\right ) \log ^5\left (x-\log ^2(x)\right )+\left (8 e^{x+4} x^2 (\log (3)+i \pi )^2 \log (x)-4 e^{x+4} x^3 (\log (3)+i \pi )^2\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{x+4} \left (2 x^4-4 x^3\right ) (\log (3)+i \pi )^2+e^{x+4} \left (4 x^2-2 x^3\right ) (\log (3)+i \pi )^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (x (\log (3)+i \pi )^4-(\log (3)+i \pi )^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )-2 (\log (3)+i \pi )^4 \log (x)+x (\log (3)+i \pi )^4}{x^5 \left (4 \log ^2(x)-4 x\right ) \log ^5\left (x-\log ^2(x)\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (4 e^{x+4} x^2 \log ^2\left (x-\log ^2(x)\right )+\pi ^2 \left (1-\frac {\log (3) (\log (3)+2 i \pi )}{\pi ^2}\right )\right ) \left (2 e^{x+4} x^4 \log ^3\left (x-\log ^2(x)\right )-2 e^{x+4} x^3 \log ^2(x) \log ^3\left (x-\log ^2(x)\right )-\pi ^2 x \left (1-\frac {\log (3) (\log (3)+2 i \pi )}{\pi ^2}\right ) \log \left (x-\log ^2(x)\right )+\pi ^2 \left (1-\frac {\log (3) (\log (3)+2 i \pi )}{\pi ^2}\right ) \log ^2(x) \log \left (x-\log ^2(x)\right )-\pi ^2 x \left (1-\frac {\log (3) (\log (3)+2 i \pi )}{\pi ^2}\right )+2 \pi ^2 \left (1-\frac {\log (3) (\log (3)+2 i \pi )}{\pi ^2}\right ) \log (x)\right )}{4 x^5 \left (x-\log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int -\frac {\left (4 e^{x+4} x^2 \log ^2\left (x-\log ^2(x)\right )+(\pi -i \log (3))^2\right ) \left (-2 e^{x+4} \log ^3\left (x-\log ^2(x)\right ) x^4+2 e^{x+4} \log ^2(x) \log ^3\left (x-\log ^2(x)\right ) x^3+(\pi -i \log (3))^2 \log \left (x-\log ^2(x)\right ) x+(\pi -i \log (3))^2 x-2 (\pi -i \log (3))^2 \log (x)-(\pi -i \log (3))^2 \log ^2(x) \log \left (x-\log ^2(x)\right )\right )}{x^5 \left (x-\log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{4} \int \frac {\left (4 e^{x+4} x^2 \log ^2\left (x-\log ^2(x)\right )+(\pi -i \log (3))^2\right ) \left (-2 e^{x+4} \log ^3\left (x-\log ^2(x)\right ) x^4+2 e^{x+4} \log ^2(x) \log ^3\left (x-\log ^2(x)\right ) x^3+(\pi -i \log (3))^2 \log \left (x-\log ^2(x)\right ) x+(\pi -i \log (3))^2 x-2 (\pi -i \log (3))^2 \log (x)-(\pi -i \log (3))^2 \log ^2(x) \log \left (x-\log ^2(x)\right )\right )}{x^5 \left (x-\log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (\frac {(\pi -i \log (3))^4 \left (-\log \left (x-\log ^2(x)\right ) \log ^2(x)-2 \log (x)+x+x \log \left (x-\log ^2(x)\right )\right )}{x^5 \left (x-\log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}-8 e^{2 x+8}-\frac {2 e^{x+4} (\pi -i \log (3))^2 \left (\log \left (x-\log ^2(x)\right ) x^2-\log ^2(x) \log \left (x-\log ^2(x)\right ) x-2 \log \left (x-\log ^2(x)\right ) x-2 x+4 \log (x)+2 \log ^2(x) \log \left (x-\log ^2(x)\right )\right )}{x^3 \left (x-\log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (\frac {(\pi -i \log (3))^4}{4 x^4 \log ^4\left (x-\log ^2(x)\right )}+\frac {2 e^{x+4} (\pi -i \log (3))^2 \left (x^2 \log \left (x-\log ^2(x)\right )-x \log ^2(x) \log \left (x-\log ^2(x)\right )\right )}{x^3 \left (x-\log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )}+4 e^{2 x+8}\right )\) |
Input:
Int[(x*(I*Pi + Log[3])^4 - 2*(I*Pi + Log[3])^4*Log[x] + (x*(I*Pi + Log[3]) ^4 - (I*Pi + Log[3])^4*Log[x]^2)*Log[x - Log[x]^2] + (-4*E^(4 + x)*x^3*(I* Pi + Log[3])^2 + 8*E^(4 + x)*x^2*(I*Pi + Log[3])^2*Log[x])*Log[x - Log[x]^ 2]^2 + (E^(4 + x)*(-4*x^3 + 2*x^4)*(I*Pi + Log[3])^2 + E^(4 + x)*(4*x^2 - 2*x^3)*(I*Pi + Log[3])^2*Log[x]^2)*Log[x - Log[x]^2]^3 + (-8*E^(8 + 2*x)*x ^6 + 8*E^(8 + 2*x)*x^5*Log[x]^2)*Log[x - Log[x]^2]^5)/((-4*x^6 + 4*x^5*Log [x]^2)*Log[x - Log[x]^2]^5),x]
Output:
(4*E^(8 + 2*x) + (Pi - I*Log[3])^4/(4*x^4*Log[x - Log[x]^2]^4) + (2*E^(4 + x)*(Pi - I*Log[3])^2*(x^2*Log[x - Log[x]^2] - x*Log[x]^2*Log[x - Log[x]^2 ]))/(x^3*(x - Log[x]^2)*Log[x - Log[x]^2]^3))/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.)) ^(p_.), x_Symbol] :> Int[u*x^(p*r)*(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (34 ) = 68\).
Time = 0.57 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.37
\[{\mathrm e}^{8+2 x}+\frac {-16 i \ln \left (3\right ) \pi \,x^{2} {\mathrm e}^{4+x} \ln \left (-\ln \left (x \right )^{2}+x \right )^{2}-8 \ln \left (3\right )^{2} x^{2} {\mathrm e}^{4+x} \ln \left (-\ln \left (x \right )^{2}+x \right )^{2}+8 \pi ^{2} x^{2} {\mathrm e}^{4+x} \ln \left (-\ln \left (x \right )^{2}+x \right )^{2}+4 i \ln \left (3\right )^{3} \pi -4 i \ln \left (3\right ) \pi ^{3}+\ln \left (3\right )^{4}-6 \ln \left (3\right )^{2} \pi ^{2}+\pi ^{4}}{16 x^{4} \ln \left (-\ln \left (x \right )^{2}+x \right )^{4}}\]
Input:
int(((8*x^5*exp(4+x)^2*ln(x)^2-8*x^6*exp(4+x)^2)*ln(-ln(x)^2+x)^5+((-2*x^3 +4*x^2)*(ln(3)+I*Pi)^2*exp(4+x)*ln(x)^2+(2*x^4-4*x^3)*(ln(3)+I*Pi)^2*exp(4 +x))*ln(-ln(x)^2+x)^3+(8*x^2*(ln(3)+I*Pi)^2*exp(4+x)*ln(x)-4*x^3*(ln(3)+I* Pi)^2*exp(4+x))*ln(-ln(x)^2+x)^2+(-(ln(3)+I*Pi)^4*ln(x)^2+x*(ln(3)+I*Pi)^4 )*ln(-ln(x)^2+x)-2*(ln(3)+I*Pi)^4*ln(x)+x*(ln(3)+I*Pi)^4)/(4*x^5*ln(x)^2-4 *x^6)/ln(-ln(x)^2+x)^5,x)
Output:
exp(8+2*x)+1/16*(-16*I*ln(3)*Pi*x^2*exp(4+x)*ln(-ln(x)^2+x)^2-8*ln(3)^2*x^ 2*exp(4+x)*ln(-ln(x)^2+x)^2+8*Pi^2*x^2*exp(4+x)*ln(-ln(x)^2+x)^2+4*I*ln(3) ^3*Pi-4*I*ln(3)*Pi^3+ln(3)^4-6*ln(3)^2*Pi^2+Pi^4)/x^4/ln(-ln(x)^2+x)^4
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (34) = 68\).
Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.92 \[ \int \frac {x (i \pi +\log (3))^4-2 (i \pi +\log (3))^4 \log (x)+\left (x (i \pi +\log (3))^4-(i \pi +\log (3))^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )+\left (-4 e^{4+x} x^3 (i \pi +\log (3))^2+8 e^{4+x} x^2 (i \pi +\log (3))^2 \log (x)\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{4+x} \left (-4 x^3+2 x^4\right ) (i \pi +\log (3))^2+e^{4+x} \left (4 x^2-2 x^3\right ) (i \pi +\log (3))^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (-8 e^{8+2 x} x^6+8 e^{8+2 x} x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}{\left (-4 x^6+4 x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx=\frac {16 \, x^{4} e^{\left (2 \, x + 8\right )} \log \left (-\log \left (x\right )^{2} + x\right )^{4} + \pi ^{4} - 4 i \, \pi ^{3} \log \left (3\right ) - 6 \, \pi ^{2} \log \left (3\right )^{2} + 4 i \, \pi \log \left (3\right )^{3} + \log \left (3\right )^{4} + 8 \, {\left (\pi ^{2} x^{2} - 2 i \, \pi x^{2} \log \left (3\right ) - x^{2} \log \left (3\right )^{2}\right )} e^{\left (x + 4\right )} \log \left (-\log \left (x\right )^{2} + x\right )^{2}}{16 \, x^{4} \log \left (-\log \left (x\right )^{2} + x\right )^{4}} \] Input:
integrate(((8*x^5*exp(4+x)^2*log(x)^2-8*x^6*exp(4+x)^2)*log(-log(x)^2+x)^5 +((-2*x^3+4*x^2)*(log(3)+I*pi)^2*exp(4+x)*log(x)^2+(2*x^4-4*x^3)*(log(3)+I *pi)^2*exp(4+x))*log(-log(x)^2+x)^3+(8*x^2*(log(3)+I*pi)^2*exp(4+x)*log(x) -4*x^3*(log(3)+I*pi)^2*exp(4+x))*log(-log(x)^2+x)^2+(-(log(3)+I*pi)^4*log( x)^2+x*(log(3)+I*pi)^4)*log(-log(x)^2+x)-2*(log(3)+I*pi)^4*log(x)+x*(log(3 )+I*pi)^4)/(4*x^5*log(x)^2-4*x^6)/log(-log(x)^2+x)^5,x, algorithm="fricas" )
Output:
1/16*(16*x^4*e^(2*x + 8)*log(-log(x)^2 + x)^4 + pi^4 - 4*I*pi^3*log(3) - 6 *pi^2*log(3)^2 + 4*I*pi*log(3)^3 + log(3)^4 + 8*(pi^2*x^2 - 2*I*pi*x^2*log (3) - x^2*log(3)^2)*e^(x + 4)*log(-log(x)^2 + x)^2)/(x^4*log(-log(x)^2 + x )^4)
Timed out. \[ \int \frac {x (i \pi +\log (3))^4-2 (i \pi +\log (3))^4 \log (x)+\left (x (i \pi +\log (3))^4-(i \pi +\log (3))^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )+\left (-4 e^{4+x} x^3 (i \pi +\log (3))^2+8 e^{4+x} x^2 (i \pi +\log (3))^2 \log (x)\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{4+x} \left (-4 x^3+2 x^4\right ) (i \pi +\log (3))^2+e^{4+x} \left (4 x^2-2 x^3\right ) (i \pi +\log (3))^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (-8 e^{8+2 x} x^6+8 e^{8+2 x} x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}{\left (-4 x^6+4 x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx=\text {Timed out} \] Input:
integrate(((8*x**5*exp(4+x)**2*ln(x)**2-8*x**6*exp(4+x)**2)*ln(-ln(x)**2+x )**5+((-2*x**3+4*x**2)*(ln(3)+I*pi)**2*exp(4+x)*ln(x)**2+(2*x**4-4*x**3)*( ln(3)+I*pi)**2*exp(4+x))*ln(-ln(x)**2+x)**3+(8*x**2*(ln(3)+I*pi)**2*exp(4+ x)*ln(x)-4*x**3*(ln(3)+I*pi)**2*exp(4+x))*ln(-ln(x)**2+x)**2+(-(ln(3)+I*pi )**4*ln(x)**2+x*(ln(3)+I*pi)**4)*ln(-ln(x)**2+x)-2*(ln(3)+I*pi)**4*ln(x)+x *(ln(3)+I*pi)**4)/(4*x**5*ln(x)**2-4*x**6)/ln(-ln(x)**2+x)**5,x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (34) = 68\).
Time = 0.55 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.74 \[ \int \frac {x (i \pi +\log (3))^4-2 (i \pi +\log (3))^4 \log (x)+\left (x (i \pi +\log (3))^4-(i \pi +\log (3))^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )+\left (-4 e^{4+x} x^3 (i \pi +\log (3))^2+8 e^{4+x} x^2 (i \pi +\log (3))^2 \log (x)\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{4+x} \left (-4 x^3+2 x^4\right ) (i \pi +\log (3))^2+e^{4+x} \left (4 x^2-2 x^3\right ) (i \pi +\log (3))^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (-8 e^{8+2 x} x^6+8 e^{8+2 x} x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}{\left (-4 x^6+4 x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx=\frac {16 \, x^{4} e^{\left (2 \, x + 8\right )} \log \left (-\log \left (x\right )^{2} + x\right )^{4} + 8 \, {\left (\pi ^{2} - 2 i \, \pi \log \left (3\right ) - \log \left (3\right )^{2}\right )} x^{2} e^{\left (x + 4\right )} \log \left (-\log \left (x\right )^{2} + x\right )^{2} + \pi ^{4} - 4 i \, \pi ^{3} \log \left (3\right ) - 6 \, \pi ^{2} \log \left (3\right )^{2} + 4 i \, \pi \log \left (3\right )^{3} + \log \left (3\right )^{4}}{16 \, x^{4} \log \left (-\log \left (x\right )^{2} + x\right )^{4}} \] Input:
integrate(((8*x^5*exp(4+x)^2*log(x)^2-8*x^6*exp(4+x)^2)*log(-log(x)^2+x)^5 +((-2*x^3+4*x^2)*(log(3)+I*pi)^2*exp(4+x)*log(x)^2+(2*x^4-4*x^3)*(log(3)+I *pi)^2*exp(4+x))*log(-log(x)^2+x)^3+(8*x^2*(log(3)+I*pi)^2*exp(4+x)*log(x) -4*x^3*(log(3)+I*pi)^2*exp(4+x))*log(-log(x)^2+x)^2+(-(log(3)+I*pi)^4*log( x)^2+x*(log(3)+I*pi)^4)*log(-log(x)^2+x)-2*(log(3)+I*pi)^4*log(x)+x*(log(3 )+I*pi)^4)/(4*x^5*log(x)^2-4*x^6)/log(-log(x)^2+x)^5,x, algorithm="maxima" )
Output:
1/16*(16*x^4*e^(2*x + 8)*log(-log(x)^2 + x)^4 + 8*(pi^2 - 2*I*pi*log(3) - log(3)^2)*x^2*e^(x + 4)*log(-log(x)^2 + x)^2 + pi^4 - 4*I*pi^3*log(3) - 6* pi^2*log(3)^2 + 4*I*pi*log(3)^3 + log(3)^4)/(x^4*log(-log(x)^2 + x)^4)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (34) = 68\).
Time = 2.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.66 \[ \int \frac {x (i \pi +\log (3))^4-2 (i \pi +\log (3))^4 \log (x)+\left (x (i \pi +\log (3))^4-(i \pi +\log (3))^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )+\left (-4 e^{4+x} x^3 (i \pi +\log (3))^2+8 e^{4+x} x^2 (i \pi +\log (3))^2 \log (x)\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{4+x} \left (-4 x^3+2 x^4\right ) (i \pi +\log (3))^2+e^{4+x} \left (4 x^2-2 x^3\right ) (i \pi +\log (3))^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (-8 e^{8+2 x} x^6+8 e^{8+2 x} x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}{\left (-4 x^6+4 x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx=\frac {16 \, x^{4} e^{\left (2 \, x + 8\right )} \log \left (-\log \left (x\right )^{2} + x\right )^{4} + 8 \, \pi ^{2} x^{2} e^{\left (x + 4\right )} \log \left (-\log \left (x\right )^{2} + x\right )^{2} - 16 i \, \pi x^{2} e^{\left (x + 4\right )} \log \left (3\right ) \log \left (-\log \left (x\right )^{2} + x\right )^{2} - 8 \, x^{2} e^{\left (x + 4\right )} \log \left (3\right )^{2} \log \left (-\log \left (x\right )^{2} + x\right )^{2} + \pi ^{4} - 4 i \, \pi ^{3} \log \left (3\right ) - 6 \, \pi ^{2} \log \left (3\right )^{2} + 4 i \, \pi \log \left (3\right )^{3} + \log \left (3\right )^{4}}{16 \, x^{4} \log \left (-\log \left (x\right )^{2} + x\right )^{4}} \] Input:
integrate(((8*x^5*exp(4+x)^2*log(x)^2-8*x^6*exp(4+x)^2)*log(-log(x)^2+x)^5 +((-2*x^3+4*x^2)*(log(3)+I*pi)^2*exp(4+x)*log(x)^2+(2*x^4-4*x^3)*(log(3)+I *pi)^2*exp(4+x))*log(-log(x)^2+x)^3+(8*x^2*(log(3)+I*pi)^2*exp(4+x)*log(x) -4*x^3*(log(3)+I*pi)^2*exp(4+x))*log(-log(x)^2+x)^2+(-(log(3)+I*pi)^4*log( x)^2+x*(log(3)+I*pi)^4)*log(-log(x)^2+x)-2*(log(3)+I*pi)^4*log(x)+x*(log(3 )+I*pi)^4)/(4*x^5*log(x)^2-4*x^6)/log(-log(x)^2+x)^5,x, algorithm="giac")
Output:
1/16*(16*x^4*e^(2*x + 8)*log(-log(x)^2 + x)^4 + 8*pi^2*x^2*e^(x + 4)*log(- log(x)^2 + x)^2 - 16*I*pi*x^2*e^(x + 4)*log(3)*log(-log(x)^2 + x)^2 - 8*x^ 2*e^(x + 4)*log(3)^2*log(-log(x)^2 + x)^2 + pi^4 - 4*I*pi^3*log(3) - 6*pi^ 2*log(3)^2 + 4*I*pi*log(3)^3 + log(3)^4)/(x^4*log(-log(x)^2 + x)^4)
Timed out. \[ \int \frac {x (i \pi +\log (3))^4-2 (i \pi +\log (3))^4 \log (x)+\left (x (i \pi +\log (3))^4-(i \pi +\log (3))^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )+\left (-4 e^{4+x} x^3 (i \pi +\log (3))^2+8 e^{4+x} x^2 (i \pi +\log (3))^2 \log (x)\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{4+x} \left (-4 x^3+2 x^4\right ) (i \pi +\log (3))^2+e^{4+x} \left (4 x^2-2 x^3\right ) (i \pi +\log (3))^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (-8 e^{8+2 x} x^6+8 e^{8+2 x} x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}{\left (-4 x^6+4 x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx=\text {Hanged} \] Input:
int(-(2*log(x)*(Pi*1i + log(3))^4 + log(x - log(x)^2)^2*(4*x^3*exp(x + 4)* (Pi*1i + log(3))^2 - 8*x^2*exp(x + 4)*log(x)*(Pi*1i + log(3))^2) - x*(Pi*1 i + log(3))^4 + log(x - log(x)^2)^3*(exp(x + 4)*(Pi*1i + log(3))^2*(4*x^3 - 2*x^4) - exp(x + 4)*log(x)^2*(Pi*1i + log(3))^2*(4*x^2 - 2*x^3)) + log(x - log(x)^2)*(log(x)^2*(Pi*1i + log(3))^4 - x*(Pi*1i + log(3))^4) + log(x - log(x)^2)^5*(8*x^6*exp(2*x + 8) - 8*x^5*exp(2*x + 8)*log(x)^2))/(log(x - log(x)^2)^5*(4*x^5*log(x)^2 - 4*x^6)),x)
Output:
\text{Hanged}
Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 3.95 \[ \int \frac {x (i \pi +\log (3))^4-2 (i \pi +\log (3))^4 \log (x)+\left (x (i \pi +\log (3))^4-(i \pi +\log (3))^4 \log ^2(x)\right ) \log \left (x-\log ^2(x)\right )+\left (-4 e^{4+x} x^3 (i \pi +\log (3))^2+8 e^{4+x} x^2 (i \pi +\log (3))^2 \log (x)\right ) \log ^2\left (x-\log ^2(x)\right )+\left (e^{4+x} \left (-4 x^3+2 x^4\right ) (i \pi +\log (3))^2+e^{4+x} \left (4 x^2-2 x^3\right ) (i \pi +\log (3))^2 \log ^2(x)\right ) \log ^3\left (x-\log ^2(x)\right )+\left (-8 e^{8+2 x} x^6+8 e^{8+2 x} x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )}{\left (-4 x^6+4 x^5 \log ^2(x)\right ) \log ^5\left (x-\log ^2(x)\right )} \, dx=\frac {16 e^{2 x} \mathrm {log}\left (-\mathrm {log}\left (x \right )^{2}+x \right )^{4} e^{8} x^{4}-8 e^{x} \mathrm {log}\left (-\mathrm {log}\left (x \right )^{2}+x \right )^{2} \mathrm {log}\left (3\right )^{2} e^{4} x^{2}-16 e^{x} \mathrm {log}\left (-\mathrm {log}\left (x \right )^{2}+x \right )^{2} \mathrm {log}\left (3\right ) e^{4} i \pi \,x^{2}+8 e^{x} \mathrm {log}\left (-\mathrm {log}\left (x \right )^{2}+x \right )^{2} e^{4} \pi ^{2} x^{2}+\mathrm {log}\left (3\right )^{4}+4 \mathrm {log}\left (3\right )^{3} i \pi -6 \mathrm {log}\left (3\right )^{2} \pi ^{2}-4 \,\mathrm {log}\left (3\right ) i \,\pi ^{3}+\pi ^{4}}{16 \mathrm {log}\left (-\mathrm {log}\left (x \right )^{2}+x \right )^{4} x^{4}} \] Input:
int(((8*x^5*exp(4+x)^2*log(x)^2-8*x^6*exp(4+x)^2)*log(-log(x)^2+x)^5+((-2* x^3+4*x^2)*(log(3)+I*Pi)^2*exp(4+x)*log(x)^2+(2*x^4-4*x^3)*(log(3)+I*Pi)^2 *exp(4+x))*log(-log(x)^2+x)^3+(8*x^2*(log(3)+I*Pi)^2*exp(4+x)*log(x)-4*x^3 *(log(3)+I*Pi)^2*exp(4+x))*log(-log(x)^2+x)^2+(-(log(3)+I*Pi)^4*log(x)^2+x *(log(3)+I*Pi)^4)*log(-log(x)^2+x)-2*(log(3)+I*Pi)^4*log(x)+x*(log(3)+I*Pi )^4)/(4*x^5*log(x)^2-4*x^6)/log(-log(x)^2+x)^5,x)
Output:
(16*e**(2*x)*log( - log(x)**2 + x)**4*e**8*x**4 - 8*e**x*log( - log(x)**2 + x)**2*log(3)**2*e**4*x**2 - 16*e**x*log( - log(x)**2 + x)**2*log(3)*e**4 *i*pi*x**2 + 8*e**x*log( - log(x)**2 + x)**2*e**4*pi**2*x**2 + log(3)**4 + 4*log(3)**3*i*pi - 6*log(3)**2*pi**2 - 4*log(3)*i*pi**3 + pi**4)/(16*log( - log(x)**2 + x)**4*x**4)