Integrand size = 104, antiderivative size = 31 \[ \int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx=\frac {e^{-\frac {e^x}{\log (2)}} x \left (x+\log \left ((3+x)^2\right )\right )}{4 (11-x)} \] Output:
1/4*(ln((3+x)^2)+x)*x/(11-x)/exp(exp(x)/ln(2))
\[ \int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx=\int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx \] Input:
Integrate[(E^x*(-33*x^2 - 8*x^3 + x^4) + (88*x + 17*x^2 - x^3)*Log[2] + (E ^x*(-33*x - 8*x^2 + x^3) + (33 + 11*x)*Log[2])*Log[9 + 6*x + x^2])/(E^(E^x /Log[2])*(1452 + 220*x - 76*x^2 + 4*x^3)*Log[2]),x]
Output:
Integrate[(E^x*(-33*x^2 - 8*x^3 + x^4) + (88*x + 17*x^2 - x^3)*Log[2] + (E ^x*(-33*x - 8*x^2 + x^3) + (33 + 11*x)*Log[2])*Log[9 + 6*x + x^2])/(E^(E^x /Log[2])*(1452 + 220*x - 76*x^2 + 4*x^3)), x]/Log[2]
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.
| \(\displaystyle \int \frac {e^{-\frac {e^x}{\log (2)}} \left (\left (-x^3+17 x^2+88 x\right ) \log (2)+\left (e^x \left (x^3-8 x^2-33 x\right )+(11 x+33) \log (2)\right ) \log \left (x^2+6 x+9\right )+e^x \left (x^4-8 x^3-33 x^2\right )\right )}{\left (4 x^3-76 x^2+220 x+1452\right ) \log (2)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {e^{-\frac {e^x}{\log (2)}} \left (-\log (2) \left (-x^3+17 x^2+88 x\right )+e^x \left (-x^4+8 x^3+33 x^2\right )+\left (e^x \left (-x^3+8 x^2+33 x\right )-11 (x+3) \log (2)\right ) \log \left (x^2+6 x+9\right )\right )}{4 \left (x^3-19 x^2+55 x+363\right )}dx}{\log (2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {e^{-\frac {e^x}{\log (2)}} \left (-\log (2) \left (-x^3+17 x^2+88 x\right )+e^x \left (-x^4+8 x^3+33 x^2\right )+\left (e^x \left (-x^3+8 x^2+33 x\right )-11 (x+3) \log (2)\right ) \log \left (x^2+6 x+9\right )\right )}{x^3-19 x^2+55 x+363}dx}{4 \log (2)}\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle -\frac {\int \left (-\frac {e^{-\frac {e^x}{\log (2)}} \left (-\log (2) \left (-x^3+17 x^2+88 x\right )+e^x \left (-x^4+8 x^3+33 x^2\right )+\left (e^x \left (-x^3+8 x^2+33 x\right )-11 (x+3) \log (2)\right ) \log \left (x^2+6 x+9\right )\right )}{196 (x-11)}+\frac {e^{-\frac {e^x}{\log (2)}} \left (-\log (2) \left (-x^3+17 x^2+88 x\right )+e^x \left (-x^4+8 x^3+33 x^2\right )+\left (e^x \left (-x^3+8 x^2+33 x\right )-11 (x+3) \log (2)\right ) \log \left (x^2+6 x+9\right )\right )}{196 (x+3)}+\frac {e^{-\frac {e^x}{\log (2)}} \left (-\log (2) \left (-x^3+17 x^2+88 x\right )+e^x \left (-x^4+8 x^3+33 x^2\right )+\left (e^x \left (-x^3+8 x^2+33 x\right )-11 (x+3) \log (2)\right ) \log \left (x^2+6 x+9\right )\right )}{14 (x-11)^2}\right )dx}{4 \log (2)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-11 \log (2) \log \left ((x+3)^2\right ) \int \frac {e^{-\frac {e^x}{\log (2)}}}{(x-11)^2}dx-121 \log (2) \int \frac {e^{-\frac {e^x}{\log (2)}}}{(x-11)^2}dx-11 \log \left ((x+3)^2\right ) \int \frac {e^{x-\frac {e^x}{\log (2)}}}{x-11}dx-121 \int \frac {e^{x-\frac {e^x}{\log (2)}}}{x-11}dx+\frac {11}{7} \log (2) \int \frac {e^{-\frac {e^x}{\log (2)}}}{x-11}dx-\int e^{x-\frac {e^x}{\log (2)}} xdx-\frac {11}{7} \log (2) \int \frac {e^{-\frac {e^x}{\log (2)}}}{x+3}dx+22 \log (2) \int \frac {\int \frac {e^{-\frac {e^x}{\log (2)}}}{(x-11)^2}dx}{x+3}dx+22 \int \frac {\int \frac {e^{x-\frac {e^x}{\log (2)}}}{x-11}dx}{x+3}dx+\log (2) \operatorname {ExpIntegralEi}\left (-\frac {e^x}{\log (2)}\right )+\log (2) e^{-\frac {e^x}{\log (2)}} \log \left ((x+3)^2\right )+11 \log (2) e^{-\frac {e^x}{\log (2)}}}{4 \log (2)}\) |
Input:
Int[(E^x*(-33*x^2 - 8*x^3 + x^4) + (88*x + 17*x^2 - x^3)*Log[2] + (E^x*(-3 3*x - 8*x^2 + x^3) + (33 + 11*x)*Log[2])*Log[9 + 6*x + x^2])/(E^(E^x/Log[2 ])*(1452 + 220*x - 76*x^2 + 4*x^3)*Log[2]),x]
Output:
$Aborted
Time = 2.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35
| method | result | size |
| parallelrisch | \(-\frac {\left (x^{2} \ln \left (2\right )+\ln \left (2\right ) \ln \left (x^{2}+6 x +9\right ) x \right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{\ln \left (2\right )}}}{4 \ln \left (2\right ) \left (x -11\right )}\) | \(42\) |
| risch | \(-\frac {x \left (-i \pi \operatorname {csgn}\left (i \left (3+x \right )\right )^{2} \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (3+x \right )\right ) \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (3+x \right )^{2}\right )^{3}+2 x +4 \ln \left (3+x \right )\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{\ln \left (2\right )}}}{8 \left (x -11\right )}\) | \(87\) |
Input:
int((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*ln(2))*ln(x^2+6*x+9)+(x^4-8*x^3-33 *x^2)*exp(x)+(-x^3+17*x^2+88*x)*ln(2))/(4*x^3-76*x^2+220*x+1452)/ln(2)/exp (exp(x)/ln(2)),x,method=_RETURNVERBOSE)
Output:
-1/4/ln(2)*(x^2*ln(2)+ln(2)*ln(x^2+6*x+9)*x)/(x-11)/exp(exp(x)/ln(2))
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx=-\frac {{\left (x^{2} + x \log \left (x^{2} + 6 \, x + 9\right )\right )} e^{\left (-\frac {e^{x}}{\log \left (2\right )}\right )}}{4 \, {\left (x - 11\right )}} \] Input:
integrate((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*log(2))*log(x^2+6*x+9)+(x^4- 8*x^3-33*x^2)*exp(x)+(-x^3+17*x^2+88*x)*log(2))/(4*x^3-76*x^2+220*x+1452)/ log(2)/exp(exp(x)/log(2)),x, algorithm="fricas")
Output:
-1/4*(x^2 + x*log(x^2 + 6*x + 9))*e^(-e^x/log(2))/(x - 11)
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx=\frac {\left (- x^{2} - x \log {\left (x^{2} + 6 x + 9 \right )}\right ) e^{- \frac {e^{x}}{\log {\left (2 \right )}}}}{4 x - 44} \] Input:
integrate((((x**3-8*x**2-33*x)*exp(x)+(11*x+33)*ln(2))*ln(x**2+6*x+9)+(x** 4-8*x**3-33*x**2)*exp(x)+(-x**3+17*x**2+88*x)*ln(2))/(4*x**3-76*x**2+220*x +1452)/ln(2)/exp(exp(x)/ln(2)),x)
Output:
(-x**2 - x*log(x**2 + 6*x + 9))*exp(-exp(x)/log(2))/(4*x - 44)
Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx=-\frac {{\left (x^{2} \log \left (2\right ) + 2 \, x \log \left (2\right ) \log \left (x + 3\right )\right )} e^{\left (-\frac {e^{x}}{\log \left (2\right )}\right )}}{4 \, {\left (x - 11\right )} \log \left (2\right )} \] Input:
integrate((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*log(2))*log(x^2+6*x+9)+(x^4- 8*x^3-33*x^2)*exp(x)+(-x^3+17*x^2+88*x)*log(2))/(4*x^3-76*x^2+220*x+1452)/ log(2)/exp(exp(x)/log(2)),x, algorithm="maxima")
Output:
-1/4*(x^2*log(2) + 2*x*log(2)*log(x + 3))*e^(-e^x/log(2))/((x - 11)*log(2) )
\[ \int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx=\int { \frac {{\left ({\left (x^{4} - 8 \, x^{3} - 33 \, x^{2}\right )} e^{x} - {\left (x^{3} - 17 \, x^{2} - 88 \, x\right )} \log \left (2\right ) + {\left ({\left (x^{3} - 8 \, x^{2} - 33 \, x\right )} e^{x} + 11 \, {\left (x + 3\right )} \log \left (2\right )\right )} \log \left (x^{2} + 6 \, x + 9\right )\right )} e^{\left (-\frac {e^{x}}{\log \left (2\right )}\right )}}{4 \, {\left (x^{3} - 19 \, x^{2} + 55 \, x + 363\right )} \log \left (2\right )} \,d x } \] Input:
integrate((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*log(2))*log(x^2+6*x+9)+(x^4- 8*x^3-33*x^2)*exp(x)+(-x^3+17*x^2+88*x)*log(2))/(4*x^3-76*x^2+220*x+1452)/ log(2)/exp(exp(x)/log(2)),x, algorithm="giac")
Output:
integrate(1/4*((x^4 - 8*x^3 - 33*x^2)*e^x - (x^3 - 17*x^2 - 88*x)*log(2) + ((x^3 - 8*x^2 - 33*x)*e^x + 11*(x + 3)*log(2))*log(x^2 + 6*x + 9))*e^(-e^ x/log(2))/((x^3 - 19*x^2 + 55*x + 363)*log(2)), x)
Timed out. \[ \int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx=-\int -\frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{\ln \left (2\right )}}\,\left (\ln \left (x^2+6\,x+9\right )\,\left (\ln \left (2\right )\,\left (11\,x+33\right )-{\mathrm {e}}^x\,\left (-x^3+8\,x^2+33\,x\right )\right )-{\mathrm {e}}^x\,\left (-x^4+8\,x^3+33\,x^2\right )+\ln \left (2\right )\,\left (-x^3+17\,x^2+88\,x\right )\right )}{\ln \left (2\right )\,\left (4\,x^3-76\,x^2+220\,x+1452\right )} \,d x \] Input:
int((exp(-exp(x)/log(2))*(log(6*x + x^2 + 9)*(log(2)*(11*x + 33) - exp(x)* (33*x + 8*x^2 - x^3)) - exp(x)*(33*x^2 + 8*x^3 - x^4) + log(2)*(88*x + 17* x^2 - x^3)))/(log(2)*(220*x - 76*x^2 + 4*x^3 + 1452)),x)
Output:
-int(-(exp(-exp(x)/log(2))*(log(6*x + x^2 + 9)*(log(2)*(11*x + 33) - exp(x )*(33*x + 8*x^2 - x^3)) - exp(x)*(33*x^2 + 8*x^3 - x^4) + log(2)*(88*x + 1 7*x^2 - x^3)))/(log(2)*(220*x - 76*x^2 + 4*x^3 + 1452)), x)
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {e^x}{\log (2)}} \left (e^x \left (-33 x^2-8 x^3+x^4\right )+\left (88 x+17 x^2-x^3\right ) \log (2)+\left (e^x \left (-33 x-8 x^2+x^3\right )+(33+11 x) \log (2)\right ) \log \left (9+6 x+x^2\right )\right )}{\left (1452+220 x-76 x^2+4 x^3\right ) \log (2)} \, dx=-\frac {x \left (\mathrm {log}\left (x^{2}+6 x +9\right )+x \right )}{4 e^{\frac {e^{x}}{\mathrm {log}\left (2\right )}} \left (x -11\right )} \] Input:
int((((x^3-8*x^2-33*x)*exp(x)+(11*x+33)*log(2))*log(x^2+6*x+9)+(x^4-8*x^3- 33*x^2)*exp(x)+(-x^3+17*x^2+88*x)*log(2))/(4*x^3-76*x^2+220*x+1452)/log(2) /exp(exp(x)/log(2)),x)
Output:
( - x*(log(x**2 + 6*x + 9) + x))/(4*e**(e**x/log(2))*(x - 11))