Integrand size = 126, antiderivative size = 26 \[ \int \frac {\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right )+\left (\left (-x-e^x x\right ) \log (x)+\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (-1+e^x (1+x)\right ) \log ^2\left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx=-4+\frac {x}{\log \left (-1+e^{-x}-x\right ) \log \left (\frac {3}{\log (x)}\right )} \]
Time = 1.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right )+\left (\left (-x-e^x x\right ) \log (x)+\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (-1+e^x (1+x)\right ) \log ^2\left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx=\frac {x}{\log \left (-1+e^{-x}-x\right ) \log \left (\frac {3}{\log (x)}\right )} \]
Integrate[((-1 + E^x*(1 + x))*Log[(1 + E^x*(-1 - x))/E^x] + ((-x - E^x*x)* Log[x] + (-1 + E^x*(1 + x))*Log[(1 + E^x*(-1 - x))/E^x]*Log[x])*Log[3/Log[ x]])/((-1 + E^x*(1 + x))*Log[(1 + E^x*(-1 - x))/E^x]^2*Log[x]*Log[3/Log[x] ]^2),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.
| \(\displaystyle \int \frac {\left (e^x (x+1)-1\right ) \log \left (e^{-x} \left (e^x (-x-1)+1\right )\right )+\left (\left (-e^x x-x\right ) \log (x)+\left (e^x (x+1)-1\right ) \log \left (e^{-x} \left (e^x (-x-1)+1\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (e^x (x+1)-1\right ) \log ^2\left (e^{-x} \left (e^x (-x-1)+1\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\log \left (-x+e^{-x}-1\right ) \left (\log \left (\frac {3}{\log (x)}\right )+\frac {1}{\log (x)}\right )-\frac {\left (e^x+1\right ) x \log \left (\frac {3}{\log (x)}\right )}{e^x (x+1)-1}}{\log ^2\left (-x+e^{-x}-1\right ) \log ^2\left (\frac {3}{\log (x)}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x \log \left (-x+e^{-x}-1\right )+x \log (x) \log \left (\frac {3}{\log (x)}\right ) \log \left (-x+e^{-x}-1\right )+\log (x) \log \left (\frac {3}{\log (x)}\right ) \log \left (-x+e^{-x}-1\right )+\log \left (-x+e^{-x}-1\right )-x \log (x) \log \left (\frac {3}{\log (x)}\right )}{(x+1) \log ^2\left (-x+e^{-x}-1\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )}-\frac {x (x+2)}{(x+1) \left (e^x x+e^x-1\right ) \log ^2\left (-x+e^{-x}-1\right ) \log \left (\frac {3}{\log (x)}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {1}{\log \left (-x+e^{-x}-1\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )}dx-\int \frac {1}{\log ^2\left (-x+e^{-x}-1\right ) \log \left (\frac {3}{\log (x)}\right )}dx+\int \frac {1}{(x+1) \log ^2\left (-x+e^{-x}-1\right ) \log \left (\frac {3}{\log (x)}\right )}dx-\int \frac {1}{\left (e^x x+e^x-1\right ) \log ^2\left (-x+e^{-x}-1\right ) \log \left (\frac {3}{\log (x)}\right )}dx-\int \frac {x}{\left (e^x x+e^x-1\right ) \log ^2\left (-x+e^{-x}-1\right ) \log \left (\frac {3}{\log (x)}\right )}dx+\int \frac {1}{(x+1) \left (e^x x+e^x-1\right ) \log ^2\left (-x+e^{-x}-1\right ) \log \left (\frac {3}{\log (x)}\right )}dx+\int \frac {1}{\log \left (-x+e^{-x}-1\right ) \log \left (\frac {3}{\log (x)}\right )}dx\) |
Int[((-1 + E^x*(1 + x))*Log[(1 + E^x*(-1 - x))/E^x] + ((-x - E^x*x)*Log[x] + (-1 + E^x*(1 + x))*Log[(1 + E^x*(-1 - x))/E^x]*Log[x])*Log[3/Log[x]])/( (-1 + E^x*(1 + x))*Log[(1 + E^x*(-1 - x))/E^x]^2*Log[x]*Log[3/Log[x]]^2),x ]
3.7.16.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 122.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15
| method | result | size |
| parallelrisch | \(\frac {x}{\ln \left (\left (\left (-1-x \right ) {\mathrm e}^{x}+1\right ) {\mathrm e}^{-x}\right ) \ln \left (\frac {3}{\ln \left (x \right )}\right )}\) | \(30\) |
| risch | \(\frac {4 x}{\left (2 i \ln \left (3\right )-2 i \ln \left (\ln \left (x \right )\right )\right ) \left (\pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )^{2}-\pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )+\pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )^{3}-2 \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )^{2}+\pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )+2 \pi -2 i \ln \left ({\mathrm e}^{x} x +{\mathrm e}^{x}-1\right )+2 i \ln \left ({\mathrm e}^{x}\right )\right )}\) | \(182\) |
int(((((1+x)*exp(x)-1)*ln(x)*ln(((-1-x)*exp(x)+1)/exp(x))+(-exp(x)*x-x)*ln (x))*ln(3/ln(x))+((1+x)*exp(x)-1)*ln(((-1-x)*exp(x)+1)/exp(x)))/((1+x)*exp (x)-1)/ln(x)/ln(((-1-x)*exp(x)+1)/exp(x))^2/ln(3/ln(x))^2,x,method=_RETURN VERBOSE)
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right )+\left (\left (-x-e^x x\right ) \log (x)+\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (-1+e^x (1+x)\right ) \log ^2\left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx=\frac {x}{\log \left (-{\left ({\left (x + 1\right )} e^{x} - 1\right )} e^{\left (-x\right )}\right ) \log \left (\frac {3}{\log \left (x\right )}\right )} \]
integrate(((((1+x)*exp(x)-1)*log(x)*log(((-1-x)*exp(x)+1)/exp(x))+(-exp(x) *x-x)*log(x))*log(3/log(x))+((1+x)*exp(x)-1)*log(((-1-x)*exp(x)+1)/exp(x)) )/((1+x)*exp(x)-1)/log(x)/log(((-1-x)*exp(x)+1)/exp(x))^2/log(3/log(x))^2, x, algorithm=\
Time = 0.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right )+\left (\left (-x-e^x x\right ) \log (x)+\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (-1+e^x (1+x)\right ) \log ^2\left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx=\frac {x}{\log {\left (\left (\left (- x - 1\right ) e^{x} + 1\right ) e^{- x} \right )} \log {\left (\frac {3}{\log {\left (x \right )}} \right )}} \]
integrate(((((1+x)*exp(x)-1)*ln(x)*ln(((-1-x)*exp(x)+1)/exp(x))+(-exp(x)*x -x)*ln(x))*ln(3/ln(x))+((1+x)*exp(x)-1)*ln(((-1-x)*exp(x)+1)/exp(x)))/((1+ x)*exp(x)-1)/ln(x)/ln(((-1-x)*exp(x)+1)/exp(x))**2/ln(3/ln(x))**2,x)
Time = 0.36 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right )+\left (\left (-x-e^x x\right ) \log (x)+\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (-1+e^x (1+x)\right ) \log ^2\left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx=-\frac {x}{x \log \left (3\right ) - {\left (\log \left (3\right ) - \log \left (\log \left (x\right )\right )\right )} \log \left (-{\left (x + 1\right )} e^{x} + 1\right ) - x \log \left (\log \left (x\right )\right )} \]
integrate(((((1+x)*exp(x)-1)*log(x)*log(((-1-x)*exp(x)+1)/exp(x))+(-exp(x) *x-x)*log(x))*log(3/log(x))+((1+x)*exp(x)-1)*log(((-1-x)*exp(x)+1)/exp(x)) )/((1+x)*exp(x)-1)/log(x)/log(((-1-x)*exp(x)+1)/exp(x))^2/log(3/log(x))^2, x, algorithm=\
Time = 0.43 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right )+\left (\left (-x-e^x x\right ) \log (x)+\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (-1+e^x (1+x)\right ) \log ^2\left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx=\frac {x}{\log \left (3\right ) \log \left (-{\left (x e^{x} + e^{x} - 1\right )} e^{\left (-x\right )}\right ) - \log \left (-{\left (x e^{x} + e^{x} - 1\right )} e^{\left (-x\right )}\right ) \log \left (\log \left (x\right )\right )} \]
integrate(((((1+x)*exp(x)-1)*log(x)*log(((-1-x)*exp(x)+1)/exp(x))+(-exp(x) *x-x)*log(x))*log(3/log(x))+((1+x)*exp(x)-1)*log(((-1-x)*exp(x)+1)/exp(x)) )/((1+x)*exp(x)-1)/log(x)/log(((-1-x)*exp(x)+1)/exp(x))^2/log(3/log(x))^2, x, algorithm=\
Time = 11.07 (sec) , antiderivative size = 1156, normalized size of antiderivative = 44.46 \[ \int \frac {\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right )+\left (\left (-x-e^x x\right ) \log (x)+\left (-1+e^x (1+x)\right ) \log \left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x)\right ) \log \left (\frac {3}{\log (x)}\right )}{\left (-1+e^x (1+x)\right ) \log ^2\left (e^{-x} \left (1+e^x (-1-x)\right )\right ) \log (x) \log ^2\left (\frac {3}{\log (x)}\right )} \, dx=\text {Too large to display} \]
int((log(-exp(-x)*(exp(x)*(x + 1) - 1))*(exp(x)*(x + 1) - 1) - log(3/log(x ))*(log(x)*(x + x*exp(x)) - log(-exp(-x)*(exp(x)*(x + 1) - 1))*log(x)*(exp (x)*(x + 1) - 1)))/(log(3/log(x))^2*log(-exp(-x)*(exp(x)*(x + 1) - 1))^2*l og(x)*(exp(x)*(x + 1) - 1)),x)
((exp(2*x) - log(x) + x*exp(2*x) + exp(2*x)*log(x) + x*exp(x) - 2*x*exp(x) *log(x) - x^2*exp(x)*log(x) - 1)/(2*log(x)*(exp(x) + 1)^2) + (log(3/log(x) )*(exp(x) - exp(3*x) - exp(2*x) - 2*x*exp(2*x) - x*exp(3*x) - x*exp(x) + x ^2*exp(2*x)*log(x) - 4*x^2*exp(x)*log(x)^2 - x^3*exp(x)*log(x)^2 + 3*x*exp (x)*log(x) + x^3*exp(2*x)*log(x)^2 + 4*x*exp(2*x)*log(x) + x*exp(3*x)*log( x) + x^2*exp(x)*log(x) + 1))/(2*log(x)*(exp(x) + 1)^3) - (x*log(3/log(x))^ 2*log(x)*(4*exp(2*x) + exp(3*x) + 3*exp(x) + x*exp(2*x) + 3*exp(x)*log(x) + 4*exp(2*x)*log(x) + exp(3*x)*log(x) + x*exp(x) - x^2*exp(2*x)*log(x) + 5 *x*exp(x)*log(x) + x*exp(2*x)*log(x) + x^2*exp(x)*log(x)))/(2*(exp(x) + 1) ^3))/log(3/log(x)) - log(x)/6 + ((exp(x) + x*exp(x) - 1)/(log(x)*(exp(x) + 1)) + (log(3/log(x))*(exp(2*x)*log(x) - log(x) - x*exp(2*x) - exp(2*x) - x*exp(x) + 4*x*exp(x)*log(x) + 2*x*exp(2*x)*log(x) + x^2*exp(x)*log(x) + 1 ))/(2*log(x)*(exp(x) + 1)^2) + (x*log(3/log(x))^2*exp(x)*log(x)*(x + exp(x ) + 3))/(2*(exp(x) + 1)^2))/log(3/log(x))^2 - log(3/log(x))*((4*x + 2*x^2) /(2*x + 2*x*exp(x)) - (6*x + 3*x^2)/(2*x + 2*x*exp(x)) + log(x)*((4*x^2*ex p(x) + 2*x^2*exp(2*x) + 2*x^2)/(2*x + 2*x*exp(2*x) + 4*x*exp(x)) - (6*x^2* exp(x) + 3*x^2*exp(2*x) + 3*x^2)/(2*x + 2*x*exp(2*x) + 4*x*exp(x)) + (x*(2 *x + 2*x^2) - 4*x^2 - 2*x^3 + x*exp(x)*(2*x + 2*x^2))/(2*x + 2*x*exp(2*x) + 4*x*exp(x)) - (x*(3*x + 3*x^2) - 6*x^2 - 3*x^3 + x*exp(x)*(3*x + 3*x^2)) /(2*x + 2*x*exp(2*x) + 4*x*exp(x))) - (2*x^2*exp(x) + 2*x^2)/(2*x + 2*x...