Integrand size = 116, antiderivative size = 32 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=-1+x+\frac {1}{4} \left (5-e^{2+x}+x\right )^2+\frac {x}{\log \left (3-e^2+x\right )} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.41 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {1}{4} e^{4+2 x}+\frac {7 x}{2}+\frac {x^2}{4}+\frac {1}{2} e^x \left (-5 e^2-e^2 x\right )-\operatorname {ExpIntegralEi}\left (\log \left (3-e^2+x\right )\right )+\frac {x}{\log \left (3-e^2+x\right )}+\operatorname {LogIntegral}\left (3-e^2+x\right ) \]
Integrate[(2*x + (-6 + 2*E^2 - 2*x)*Log[3 - E^2 + x] + (-21 + E^(4 + 2*x)* (-3 + E^2 - x) - 10*x - x^2 + E^2*(7 + x) + E^(2 + x)*(18 + E^2*(-6 - x) + 9*x + x^2))*Log[3 - E^2 + x]^2)/((-6 + 2*E^2 - 2*x)*Log[3 - E^2 + x]^2),x ]
E^(4 + 2*x)/4 + (7*x)/2 + x^2/4 + (E^x*(-5*E^2 - E^2*x))/2 - ExpIntegralEi [Log[3 - E^2 + x]] + x/Log[3 - E^2 + x] + LogIntegral[3 - E^2 + x]
Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(32)=64\).
Time = 0.69 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.66, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7239, 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.
\(\displaystyle \int \frac {\left (-x^2+e^{x+2} \left (x^2+9 x+e^2 (-x-6)+18\right )-10 x+e^{2 x+4} \left (-x+e^2-3\right )+e^2 (x+7)-21\right ) \log ^2\left (x-e^2+3\right )+2 x+\left (-2 x+2 e^2-6\right ) \log \left (x-e^2+3\right )}{\left (-2 x+2 e^2-6\right ) \log ^2\left (x-e^2+3\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\frac {1}{2} \left (x+e^{2 x+4}-e^{x+2} (x+6)+7\right )+\frac {x}{\left (-x+e^2-3\right ) \log ^2\left (x-e^2+3\right )}+\frac {1}{\log \left (x-e^2+3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^2}{4}+\frac {7 x}{2}+\frac {e^{x+2}}{2}+\frac {1}{4} e^{2 x+4}-\frac {1}{2} e^{x+2} (x+6)+\frac {x-e^2+3}{\log \left (x-e^2+3\right )}-\frac {3-e^2}{\log \left (x-e^2+3\right )}\) |
Int[(2*x + (-6 + 2*E^2 - 2*x)*Log[3 - E^2 + x] + (-21 + E^(4 + 2*x)*(-3 + E^2 - x) - 10*x - x^2 + E^2*(7 + x) + E^(2 + x)*(18 + E^2*(-6 - x) + 9*x + x^2))*Log[3 - E^2 + x]^2)/((-6 + 2*E^2 - 2*x)*Log[3 - E^2 + x]^2),x]
E^(2 + x)/2 + E^(4 + 2*x)/4 + (7*x)/2 + x^2/4 - (E^(2 + x)*(6 + x))/2 - (3 - E^2)/Log[3 - E^2 + x] + (3 - E^2 + x)/Log[3 - E^2 + x]
3.17.71.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\frac {7 x}{2}+\frac {x^{2}}{4}+\frac {{\mathrm e}^{4+2 x}}{4}-\frac {5 \,{\mathrm e}^{2+x}}{2}-\frac {x \,{\mathrm e}^{2+x}}{2}+\frac {x}{\ln \left (-{\mathrm e}^{2}+3+x \right )}\) | \(43\) |
default | \(\frac {7 x}{2}+\frac {x^{2}}{4}+\frac {{\mathrm e}^{4+2 x}}{4}-\frac {{\mathrm e}^{2+x} \left (2+x \right )}{2}-\frac {3 \,{\mathrm e}^{2+x}}{2}+\frac {-{\mathrm e}^{2}+3+x}{\ln \left (-{\mathrm e}^{2}+3+x \right )}+\frac {{\mathrm e}^{2}}{\ln \left (-{\mathrm e}^{2}+3+x \right )}-\frac {3}{\ln \left (-{\mathrm e}^{2}+3+x \right )}\) | \(76\) |
parts | \(\frac {7 x}{2}+\frac {x^{2}}{4}+\frac {{\mathrm e}^{4+2 x}}{4}-\frac {{\mathrm e}^{2+x} \left (2+x \right )}{2}-\frac {3 \,{\mathrm e}^{2+x}}{2}+\frac {-{\mathrm e}^{2}+3+x}{\ln \left (-{\mathrm e}^{2}+3+x \right )}+\frac {{\mathrm e}^{2}}{\ln \left (-{\mathrm e}^{2}+3+x \right )}-\frac {3}{\ln \left (-{\mathrm e}^{2}+3+x \right )}\) | \(76\) |
parallelrisch | \(\frac {-{\mathrm e}^{4} \ln \left (-{\mathrm e}^{2}+3+x \right )+x^{2} \ln \left (-{\mathrm e}^{2}+3+x \right )-2 \,{\mathrm e}^{2+x} x \ln \left (-{\mathrm e}^{2}+3+x \right )+{\mathrm e}^{4+2 x} \ln \left (-{\mathrm e}^{2}+3+x \right )+34 \ln \left (-{\mathrm e}^{2}+3+x \right ) {\mathrm e}^{2}+14 \ln \left (-{\mathrm e}^{2}+3+x \right ) x -10 \,{\mathrm e}^{2+x} \ln \left (-{\mathrm e}^{2}+3+x \right )+4 x -93 \ln \left (-{\mathrm e}^{2}+3+x \right )}{4 \ln \left (-{\mathrm e}^{2}+3+x \right )}\) | \(120\) |
int((((exp(2)-3-x)*exp(2+x)^2+((-x-6)*exp(2)+x^2+9*x+18)*exp(2+x)+(x+7)*ex p(2)-x^2-10*x-21)*ln(-exp(2)+3+x)^2+(2*exp(2)-2*x-6)*ln(-exp(2)+3+x)+2*x)/ (2*exp(2)-2*x-6)/ln(-exp(2)+3+x)^2,x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {{\left (x^{2} - 2 \, {\left (x + 5\right )} e^{\left (x + 2\right )} + 14 \, x + e^{\left (2 \, x + 4\right )}\right )} \log \left (x - e^{2} + 3\right ) + 4 \, x}{4 \, \log \left (x - e^{2} + 3\right )} \]
integrate((((exp(2)-3-x)*exp(2+x)^2+((-x-6)*exp(2)+x^2+9*x+18)*exp(2+x)+(x +7)*exp(2)-x^2-10*x-21)*log(-exp(2)+3+x)^2+(2*exp(2)-2*x-6)*log(-exp(2)+3+ x)+2*x)/(2*exp(2)-2*x-6)/log(-exp(2)+3+x)^2,x, algorithm=\
1/4*((x^2 - 2*(x + 5)*e^(x + 2) + 14*x + e^(2*x + 4))*log(x - e^2 + 3) + 4 *x)/log(x - e^2 + 3)
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {x^{2}}{4} + \frac {7 x}{2} + \frac {x}{\log {\left (x - e^{2} + 3 \right )}} + \frac {\left (- 4 x - 20\right ) e^{x + 2}}{8} + \frac {e^{2 x + 4}}{4} \]
integrate((((exp(2)-3-x)*exp(2+x)**2+((-x-6)*exp(2)+x**2+9*x+18)*exp(2+x)+ (x+7)*exp(2)-x**2-10*x-21)*ln(-exp(2)+3+x)**2+(2*exp(2)-2*x-6)*ln(-exp(2)+ 3+x)+2*x)/(2*exp(2)-2*x-6)/ln(-exp(2)+3+x)**2,x)
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {{\left (x^{2} - 2 \, {\left (x e^{2} + 5 \, e^{2}\right )} e^{x} + 14 \, x + e^{\left (2 \, x + 4\right )}\right )} \log \left (x - e^{2} + 3\right ) + 4 \, x}{4 \, \log \left (x - e^{2} + 3\right )} \]
integrate((((exp(2)-3-x)*exp(2+x)^2+((-x-6)*exp(2)+x^2+9*x+18)*exp(2+x)+(x +7)*exp(2)-x^2-10*x-21)*log(-exp(2)+3+x)^2+(2*exp(2)-2*x-6)*log(-exp(2)+3+ x)+2*x)/(2*exp(2)-2*x-6)/log(-exp(2)+3+x)^2,x, algorithm=\
1/4*((x^2 - 2*(x*e^2 + 5*e^2)*e^x + 14*x + e^(2*x + 4))*log(x - e^2 + 3) + 4*x)/log(x - e^2 + 3)
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.78 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {{\left (x + 2\right )}^{2} \log \left (x - e^{2} + 3\right ) - 2 \, {\left (x + 2\right )} e^{\left (x + 2\right )} \log \left (x - e^{2} + 3\right ) + 10 \, {\left (x + 2\right )} \log \left (x - e^{2} + 3\right ) + e^{\left (2 \, x + 4\right )} \log \left (x - e^{2} + 3\right ) - 6 \, e^{\left (x + 2\right )} \log \left (x - e^{2} + 3\right ) + 4 \, x}{4 \, \log \left (x - e^{2} + 3\right )} \]
integrate((((exp(2)-3-x)*exp(2+x)^2+((-x-6)*exp(2)+x^2+9*x+18)*exp(2+x)+(x +7)*exp(2)-x^2-10*x-21)*log(-exp(2)+3+x)^2+(2*exp(2)-2*x-6)*log(-exp(2)+3+ x)+2*x)/(2*exp(2)-2*x-6)/log(-exp(2)+3+x)^2,x, algorithm=\
1/4*((x + 2)^2*log(x - e^2 + 3) - 2*(x + 2)*e^(x + 2)*log(x - e^2 + 3) + 1 0*(x + 2)*log(x - e^2 + 3) + e^(2*x + 4)*log(x - e^2 + 3) - 6*e^(x + 2)*lo g(x - e^2 + 3) + 4*x)/log(x - e^2 + 3)
Time = 10.57 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {2 x+\left (-6+2 e^2-2 x\right ) \log \left (3-e^2+x\right )+\left (-21+e^{4+2 x} \left (-3+e^2-x\right )-10 x-x^2+e^2 (7+x)+e^{2+x} \left (18+e^2 (-6-x)+9 x+x^2\right )\right ) \log ^2\left (3-e^2+x\right )}{\left (-6+2 e^2-2 x\right ) \log ^2\left (3-e^2+x\right )} \, dx=\frac {7\,x}{2}-\frac {5\,{\mathrm {e}}^{x+2}}{2}+\frac {{\mathrm {e}}^{2\,x+4}}{4}-\frac {x\,{\mathrm {e}}^{x+2}}{2}+\frac {x}{\ln \left (x-{\mathrm {e}}^2+3\right )}+\frac {x^2}{4} \]
int((log(x - exp(2) + 3)^2*(10*x - exp(x + 2)*(9*x - exp(2)*(x + 6) + x^2 + 18) + exp(2*x + 4)*(x - exp(2) + 3) - exp(2)*(x + 7) + x^2 + 21) - 2*x + log(x - exp(2) + 3)*(2*x - 2*exp(2) + 6))/(log(x - exp(2) + 3)^2*(2*x - 2 *exp(2) + 6)),x)