Integrand size = 111, antiderivative size = 27 \[ \int \frac {e^4 (14-14 x)-28 x-42 x^2+14 x^4+\left (56 x+28 x^2-28 x^3+e^4 (-14+7 x)\right ) \log (2-x)+\left (-28 x+14 x^2\right ) \log ^2(2-x)}{-2-3 x+x^3+\left (4+2 x-2 x^2\right ) \log (2-x)+(-2+x) \log ^2(2-x)} \, dx=e^4+7 x \left (x-\frac {e^4}{1+x-\log (2-x)}\right ) \] Output:
7*x*(x-exp(2)^2/(1+x-ln(2-x)))+exp(4)
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^4 (14-14 x)-28 x-42 x^2+14 x^4+\left (56 x+28 x^2-28 x^3+e^4 (-14+7 x)\right ) \log (2-x)+\left (-28 x+14 x^2\right ) \log ^2(2-x)}{-2-3 x+x^3+\left (4+2 x-2 x^2\right ) \log (2-x)+(-2+x) \log ^2(2-x)} \, dx=7 \left (x^2+\frac {e^4 x}{-1-x+\log (2-x)}\right ) \] Input:
Integrate[(E^4*(14 - 14*x) - 28*x - 42*x^2 + 14*x^4 + (56*x + 28*x^2 - 28* x^3 + E^4*(-14 + 7*x))*Log[2 - x] + (-28*x + 14*x^2)*Log[2 - x]^2)/(-2 - 3 *x + x^3 + (4 + 2*x - 2*x^2)*Log[2 - x] + (-2 + x)*Log[2 - x]^2),x]
Output:
7*(x^2 + (E^4*x)/(-1 - x + Log[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.
\(\displaystyle \int \frac {14 x^4-42 x^2+\left (14 x^2-28 x\right ) \log ^2(2-x)+\left (-28 x^3+28 x^2+56 x+e^4 (7 x-14)\right ) \log (2-x)-28 x+e^4 (14-14 x)}{x^3+\left (-2 x^2+2 x+4\right ) \log (2-x)-3 x+(x-2) \log ^2(2-x)-2} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-14 x^4+42 x^2-\left (14 x^2-28 x\right ) \log ^2(2-x)-\left (-28 x^3+28 x^2+56 x+e^4 (7 x-14)\right ) \log (2-x)+28 x-e^4 (14-14 x)}{(2-x) (x-\log (2-x)+1)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (14 x+\frac {7 e^4 (x-3) x}{(x-2) (x-\log (2-x)+1)^2}-\frac {7 e^4}{x-\log (2-x)+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -7 e^4 \int \frac {1}{(x-\log (2-x)+1)^2}dx-14 e^4 \int \frac {1}{(x-2) (x-\log (2-x)+1)^2}dx+7 e^4 \int \frac {x}{(x-\log (2-x)+1)^2}dx-7 e^4 \int \frac {1}{x-\log (2-x)+1}dx+7 x^2\) |
Input:
Int[(E^4*(14 - 14*x) - 28*x - 42*x^2 + 14*x^4 + (56*x + 28*x^2 - 28*x^3 + E^4*(-14 + 7*x))*Log[2 - x] + (-28*x + 14*x^2)*Log[2 - x]^2)/(-2 - 3*x + x ^3 + (4 + 2*x - 2*x^2)*Log[2 - x] + (-2 + x)*Log[2 - x]^2),x]
Output:
$Aborted
Time = 2.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
risch | \(7 x^{2}-\frac {7 \,{\mathrm e}^{4} x}{1+x -\ln \left (2-x \right )}\) | \(25\) |
norman | \(\frac {-7 \,{\mathrm e}^{4} \ln \left (2-x \right )+7 x^{2}+7 x^{3}-7 x^{2} \ln \left (2-x \right )+7 \,{\mathrm e}^{4}}{1+x -\ln \left (2-x \right )}\) | \(55\) |
parallelrisch | \(-\frac {7 x \,{\mathrm e}^{4}+28-7 x^{3}+7 x^{2} \ln \left (2-x \right )-7 x^{2}+28 x -28 \ln \left (2-x \right )}{1+x -\ln \left (2-x \right )}\) | \(57\) |
derivativedivides | \(-\frac {7 \left (\left ({\mathrm e}^{4}-12\right ) \left (2-x \right )+7 \left (2-x \right )^{2}-\left (2-x \right )^{3}+4 \ln \left (2-x \right ) \left (2-x \right )-\ln \left (2-x \right ) \left (2-x \right )^{2}-2 \,{\mathrm e}^{4}\right )}{\ln \left (2-x \right )-x -1}\) | \(81\) |
default | \(-\frac {7 \left (\left ({\mathrm e}^{4}-12\right ) \left (2-x \right )+7 \left (2-x \right )^{2}-\left (2-x \right )^{3}+4 \ln \left (2-x \right ) \left (2-x \right )-\ln \left (2-x \right ) \left (2-x \right )^{2}-2 \,{\mathrm e}^{4}\right )}{\ln \left (2-x \right )-x -1}\) | \(81\) |
Input:
int(((14*x^2-28*x)*ln(2-x)^2+((7*x-14)*exp(2)^2-28*x^3+28*x^2+56*x)*ln(2-x )+(-14*x+14)*exp(2)^2+14*x^4-42*x^2-28*x)/((-2+x)*ln(2-x)^2+(-2*x^2+2*x+4) *ln(2-x)+x^3-3*x-2),x,method=_RETURNVERBOSE)
Output:
7*x^2-7*exp(4)*x/(1+x-ln(2-x))
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^4 (14-14 x)-28 x-42 x^2+14 x^4+\left (56 x+28 x^2-28 x^3+e^4 (-14+7 x)\right ) \log (2-x)+\left (-28 x+14 x^2\right ) \log ^2(2-x)}{-2-3 x+x^3+\left (4+2 x-2 x^2\right ) \log (2-x)+(-2+x) \log ^2(2-x)} \, dx=\frac {7 \, {\left (x^{3} - x^{2} \log \left (-x + 2\right ) + x^{2} - x e^{4}\right )}}{x - \log \left (-x + 2\right ) + 1} \] Input:
integrate(((14*x^2-28*x)*log(2-x)^2+((7*x-14)*exp(2)^2-28*x^3+28*x^2+56*x) *log(2-x)+(-14*x+14)*exp(2)^2+14*x^4-42*x^2-28*x)/((-2+x)*log(2-x)^2+(-2*x ^2+2*x+4)*log(2-x)+x^3-3*x-2),x, algorithm="fricas")
Output:
7*(x^3 - x^2*log(-x + 2) + x^2 - x*e^4)/(x - log(-x + 2) + 1)
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {e^4 (14-14 x)-28 x-42 x^2+14 x^4+\left (56 x+28 x^2-28 x^3+e^4 (-14+7 x)\right ) \log (2-x)+\left (-28 x+14 x^2\right ) \log ^2(2-x)}{-2-3 x+x^3+\left (4+2 x-2 x^2\right ) \log (2-x)+(-2+x) \log ^2(2-x)} \, dx=7 x^{2} + \frac {7 x e^{4}}{- x + \log {\left (2 - x \right )} - 1} \] Input:
integrate(((14*x**2-28*x)*ln(2-x)**2+((7*x-14)*exp(2)**2-28*x**3+28*x**2+5 6*x)*ln(2-x)+(-14*x+14)*exp(2)**2+14*x**4-42*x**2-28*x)/((-2+x)*ln(2-x)**2 +(-2*x**2+2*x+4)*ln(2-x)+x**3-3*x-2),x)
Output:
7*x**2 + 7*x*exp(4)/(-x + log(2 - x) - 1)
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^4 (14-14 x)-28 x-42 x^2+14 x^4+\left (56 x+28 x^2-28 x^3+e^4 (-14+7 x)\right ) \log (2-x)+\left (-28 x+14 x^2\right ) \log ^2(2-x)}{-2-3 x+x^3+\left (4+2 x-2 x^2\right ) \log (2-x)+(-2+x) \log ^2(2-x)} \, dx=\frac {7 \, {\left (x^{3} - x^{2} \log \left (-x + 2\right ) + x^{2} - x e^{4}\right )}}{x - \log \left (-x + 2\right ) + 1} \] Input:
integrate(((14*x^2-28*x)*log(2-x)^2+((7*x-14)*exp(2)^2-28*x^3+28*x^2+56*x) *log(2-x)+(-14*x+14)*exp(2)^2+14*x^4-42*x^2-28*x)/((-2+x)*log(2-x)^2+(-2*x ^2+2*x+4)*log(2-x)+x^3-3*x-2),x, algorithm="maxima")
Output:
7*(x^3 - x^2*log(-x + 2) + x^2 - x*e^4)/(x - log(-x + 2) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.48 \[ \int \frac {e^4 (14-14 x)-28 x-42 x^2+14 x^4+\left (56 x+28 x^2-28 x^3+e^4 (-14+7 x)\right ) \log (2-x)+\left (-28 x+14 x^2\right ) \log ^2(2-x)}{-2-3 x+x^3+\left (4+2 x-2 x^2\right ) \log (2-x)+(-2+x) \log ^2(2-x)} \, dx=\frac {7 \, {\left ({\left (x - 2\right )}^{3} - {\left (x - 2\right )}^{2} \log \left (-x + 2\right ) + 7 \, {\left (x - 2\right )}^{2} - {\left (x - 2\right )} e^{4} - 4 \, {\left (x - 2\right )} \log \left (-x + 2\right ) + 12 \, x - 2 \, e^{4} - 24\right )}}{x - \log \left (-x + 2\right ) + 1} \] Input:
integrate(((14*x^2-28*x)*log(2-x)^2+((7*x-14)*exp(2)^2-28*x^3+28*x^2+56*x) *log(2-x)+(-14*x+14)*exp(2)^2+14*x^4-42*x^2-28*x)/((-2+x)*log(2-x)^2+(-2*x ^2+2*x+4)*log(2-x)+x^3-3*x-2),x, algorithm="giac")
Output:
7*((x - 2)^3 - (x - 2)^2*log(-x + 2) + 7*(x - 2)^2 - (x - 2)*e^4 - 4*(x - 2)*log(-x + 2) + 12*x - 2*e^4 - 24)/(x - log(-x + 2) + 1)
Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {e^4 (14-14 x)-28 x-42 x^2+14 x^4+\left (56 x+28 x^2-28 x^3+e^4 (-14+7 x)\right ) \log (2-x)+\left (-28 x+14 x^2\right ) \log ^2(2-x)}{-2-3 x+x^3+\left (4+2 x-2 x^2\right ) \log (2-x)+(-2+x) \log ^2(2-x)} \, dx=\frac {7\,\left ({\mathrm {e}}^4-{\mathrm {e}}^4\,\ln \left (2-x\right )+x^2+x^3-x^2\,\ln \left (2-x\right )\right )}{x-\ln \left (2-x\right )+1} \] Input:
int(-(28*x - log(2 - x)*(56*x + 28*x^2 - 28*x^3 + exp(4)*(7*x - 14)) + log (2 - x)^2*(28*x - 14*x^2) + 42*x^2 - 14*x^4 + exp(4)*(14*x - 14))/(log(2 - x)^2*(x - 2) - 3*x + log(2 - x)*(2*x - 2*x^2 + 4) + x^3 - 2),x)
Output:
(7*(exp(4) - exp(4)*log(2 - x) + x^2 + x^3 - x^2*log(2 - x)))/(x - log(2 - x) + 1)
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {e^4 (14-14 x)-28 x-42 x^2+14 x^4+\left (56 x+28 x^2-28 x^3+e^4 (-14+7 x)\right ) \log (2-x)+\left (-28 x+14 x^2\right ) \log ^2(2-x)}{-2-3 x+x^3+\left (4+2 x-2 x^2\right ) \log (2-x)+(-2+x) \log ^2(2-x)} \, dx=\frac {7 \,\mathrm {log}\left (-x +2\right ) e^{4}+7 \,\mathrm {log}\left (-x +2\right ) x^{2}-7 e^{4}-7 x^{3}-7 x^{2}}{\mathrm {log}\left (-x +2\right )-x -1} \] Input:
int(((14*x^2-28*x)*log(2-x)^2+((7*x-14)*exp(2)^2-28*x^3+28*x^2+56*x)*log(2 -x)+(-14*x+14)*exp(2)^2+14*x^4-42*x^2-28*x)/((-2+x)*log(2-x)^2+(-2*x^2+2*x +4)*log(2-x)+x^3-3*x-2),x)
Output:
(7*(log( - x + 2)*e**4 + log( - x + 2)*x**2 - e**4 - x**3 - x**2))/(log( - x + 2) - x - 1)