Integrand size = 116, antiderivative size = 29 \[ \int \frac {-8 x-2 x \log (4)+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx=\log \left (\frac {x}{\log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}\right ) \]
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {-8 x-2 x \log (4)+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx=\log (x)-\log \left (-4 x-x \log (4)+x (4+\log (4))+\log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )\right ) \]
Integrate[(-8*x - 2*x*Log[4] + 2*Log[3/x] + (4*x + x*Log[4])*Log[3/x]^2 + (-2 + Log[3/x]^2)*Log[(E^(-4*x - x*Log[4])*(-2 + Log[3/x]^2))/6])/((-2*x + x*Log[3/x]^2)*Log[(E^(-4*x - x*Log[4])*(-2 + Log[3/x]^2))/6]),x]
Log[x] - Log[-4*x - x*Log[4] + x*(4 + Log[4]) + Log[(-2 + Log[3/x]^2)/(6*E ^(x*(4 + Log[4])))]]
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.
| \(\displaystyle \int \frac {-8 x+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (\log ^2\left (\frac {3}{x}\right )-2\right ) \log \left (\frac {1}{6} e^{x (-\log (4))-4 x} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )+2 \log \left (\frac {3}{x}\right )-2 x \log (4)}{\left (x \log ^2\left (\frac {3}{x}\right )-2 x\right ) \log \left (\frac {1}{6} e^{x (-\log (4))-4 x} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (\log ^2\left (\frac {3}{x}\right )-2\right ) \log \left (\frac {1}{6} e^{x (-\log (4))-4 x} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )+2 \log \left (\frac {3}{x}\right )+x (-8-2 \log (4))}{\left (x \log ^2\left (\frac {3}{x}\right )-2 x\right ) \log \left (\frac {1}{6} e^{x (-\log (4))-4 x} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )}dx\) |
| \(\Big \downarrow \) 3041 |
| \(\displaystyle \int \frac {(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (\log ^2\left (\frac {3}{x}\right )-2\right ) \log \left (\frac {1}{6} e^{x (-\log (4))-4 x} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )+2 \log \left (\frac {3}{x}\right )+x (-8-2 \log (4))}{x \left (\log ^2\left (\frac {3}{x}\right )-2\right ) \log \left (\frac {1}{6} e^{x (-\log (4))-4 x} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left ((4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )\right )-\left (\log ^2\left (\frac {3}{x}\right )-2\right ) \log \left (\frac {1}{6} e^{x (-\log (4))-4 x} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )-2 \log \left (\frac {3}{x}\right )-x (-8-2 \log (4))}{x \left (2-\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{x}+\frac {-4 x \left (1+\frac {\log (2)}{2}\right ) \log ^2\left (\frac {3}{x}\right )-2 \log \left (\frac {3}{x}\right )+8 x \left (1+\frac {\log (2)}{2}\right )}{x \left (2-\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 (2+\log (2)) \int \frac {1}{\left (\log ^2\left (\frac {3}{x}\right )-2\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )}dx+2 \int \frac {\log \left (\frac {3}{x}\right )}{x \left (\log ^2\left (\frac {3}{x}\right )-2\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )}dx+2 (2+\log (2)) \int \frac {\log ^2\left (\frac {3}{x}\right )}{\left (\log ^2\left (\frac {3}{x}\right )-2\right ) \log \left (\frac {1}{6} e^{-x (4+\log (4))} \left (\log ^2\left (\frac {3}{x}\right )-2\right )\right )}dx+\log (x)\) |
Int[(-8*x - 2*x*Log[4] + 2*Log[3/x] + (4*x + x*Log[4])*Log[3/x]^2 + (-2 + Log[3/x]^2)*Log[(E^(-4*x - x*Log[4])*(-2 + Log[3/x]^2))/6])/((-2*x + x*Log [3/x]^2)*Log[(E^(-4*x - x*Log[4])*(-2 + Log[3/x]^2))/6]),x]
3.8.62.3.1 Defintions of rubi rules used
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[(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]
Time = 4.67 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
| method | result | size |
| parallelrisch | \(-\ln \left (\ln \left (\frac {\left (\ln \left (\frac {3}{x}\right )^{2}-2\right ) {\mathrm e}^{-2 x \ln \left (2\right )-4 x}}{6}\right )\right )-\ln \left (\frac {3}{x}\right )\) | \(38\) |
| risch | \(\ln \left (x \right )-\ln \left (\ln \left (4^{x} {\mathrm e}^{4 x}\right )+\frac {i \left (2 \pi \operatorname {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8+4 \ln \left (3\right )^{2}-8 \ln \left (3\right ) \ln \left (x \right )+4 \ln \left (x \right )^{2}\right )\right )^{2}+\pi \,\operatorname {csgn}\left (i \left (-8+4 \ln \left (3\right )^{2}-8 \ln \left (3\right ) \ln \left (x \right )+4 \ln \left (x \right )^{2}\right )\right ) \operatorname {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x}\right ) \operatorname {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8+4 \ln \left (3\right )^{2}-8 \ln \left (3\right ) \ln \left (x \right )+4 \ln \left (x \right )^{2}\right )\right )+\pi \,\operatorname {csgn}\left (i \left (-8+4 \ln \left (3\right )^{2}-8 \ln \left (3\right ) \ln \left (x \right )+4 \ln \left (x \right )^{2}\right )\right ) \operatorname {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8+4 \ln \left (3\right )^{2}-8 \ln \left (3\right ) \ln \left (x \right )+4 \ln \left (x \right )^{2}\right )\right )^{2}-\pi \,\operatorname {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x}\right ) \operatorname {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8+4 \ln \left (3\right )^{2}-8 \ln \left (3\right ) \ln \left (x \right )+4 \ln \left (x \right )^{2}\right )\right )^{2}+\pi \operatorname {csgn}\left (i 4^{-x} {\mathrm e}^{-4 x} \left (-8+4 \ln \left (3\right )^{2}-8 \ln \left (3\right ) \ln \left (x \right )+4 \ln \left (x \right )^{2}\right )\right )^{3}-6 i \ln \left (2\right )-2 i \ln \left (3\right )+2 i \ln \left (8-4 \ln \left (3\right )^{2}+8 \ln \left (3\right ) \ln \left (x \right )-4 \ln \left (x \right )^{2}\right )-2 \pi \right )}{2}\right )\) | \(317\) |
int(((ln(3/x)^2-2)*ln(1/6*(ln(3/x)^2-2)/exp(2*x*ln(2)+4*x))+(2*x*ln(2)+4*x )*ln(3/x)^2+2*ln(3/x)-4*x*ln(2)-8*x)/(x*ln(3/x)^2-2*x)/ln(1/6*(ln(3/x)^2-2 )/exp(2*x*ln(2)+4*x)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {-8 x-2 x \log (4)+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx=-\log \left (\frac {3}{x}\right ) - \log \left (\log \left (\frac {1}{6} \, e^{\left (-2 \, x \log \left (2\right ) - 4 \, x\right )} \log \left (\frac {3}{x}\right )^{2} - \frac {1}{3} \, e^{\left (-2 \, x \log \left (2\right ) - 4 \, x\right )}\right )\right ) \]
integrate(((log(3/x)^2-2)*log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x))+(2*x *log(2)+4*x)*log(3/x)^2+2*log(3/x)-4*x*log(2)-8*x)/(x*log(3/x)^2-2*x)/log( 1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x)),x, algorithm=\
Time = 0.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-8 x-2 x \log (4)+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx=\log {\left (x \right )} - \log {\left (\log {\left (\left (\frac {\log {\left (\frac {3}{x} \right )}^{2}}{6} - \frac {1}{3}\right ) e^{- 4 x - 2 x \log {\left (2 \right )}} \right )} \right )} \]
integrate(((ln(3/x)**2-2)*ln(1/6*(ln(3/x)**2-2)/exp(2*x*ln(2)+4*x))+(2*x*l n(2)+4*x)*ln(3/x)**2+2*ln(3/x)-4*x*ln(2)-8*x)/(x*ln(3/x)**2-2*x)/ln(1/6*(l n(3/x)**2-2)/exp(2*x*ln(2)+4*x)),x)
Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-8 x-2 x \log (4)+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx=-\log \left (-2 \, x {\left (\log \left (2\right ) + 2\right )} - \log \left (3\right ) - \log \left (2\right ) + \log \left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 2\right )\right ) + \log \left (x\right ) \]
integrate(((log(3/x)^2-2)*log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x))+(2*x *log(2)+4*x)*log(3/x)^2+2*log(3/x)-4*x*log(2)-8*x)/(x*log(3/x)^2-2*x)/log( 1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x)),x, algorithm=\
-log(-2*x*(log(2) + 2) - log(3) - log(2) + log(log(3)^2 - 2*log(3)*log(x) + log(x)^2 - 2)) + log(x)
Time = 0.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-8 x-2 x \log (4)+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx=-\log \left (2 \, x \log \left (2\right ) + 4 \, x + \log \left (3\right ) + \log \left (2\right ) - \log \left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2} - 2\right )\right ) + \log \left (x\right ) \]
integrate(((log(3/x)^2-2)*log(1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x))+(2*x *log(2)+4*x)*log(3/x)^2+2*log(3/x)-4*x*log(2)-8*x)/(x*log(3/x)^2-2*x)/log( 1/6*(log(3/x)^2-2)/exp(2*x*log(2)+4*x)),x, algorithm=\
-log(2*x*log(2) + 4*x + log(3) + log(2) - log(log(3)^2 - 2*log(3)*log(x) + log(x)^2 - 2)) + log(x)
Time = 9.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-8 x-2 x \log (4)+2 \log \left (\frac {3}{x}\right )+(4 x+x \log (4)) \log ^2\left (\frac {3}{x}\right )+\left (-2+\log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )}{\left (-2 x+x \log ^2\left (\frac {3}{x}\right )\right ) \log \left (\frac {1}{6} e^{-4 x-x \log (4)} \left (-2+\log ^2\left (\frac {3}{x}\right )\right )\right )} \, dx=\ln \left (x\right )-\ln \left (\ln \left (\frac {{\mathrm {e}}^{-4\,x}\,\left (\frac {{\ln \left (\frac {3}{x}\right )}^2}{6}-\frac {1}{3}\right )}{2^{2\,x}}\right )\right ) \]