Integrand size = 81, antiderivative size = 32 \[ \int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx=\frac {\sqrt [6]{2} e^{\frac {x}{2+x}} \log (3)}{\sqrt [6]{\frac {x}{-5+4 x^2}}} \] Output:
exp(x/(2+x)-1/6*ln(x/(8*x^2-10)))*ln(3)
\[ \int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx=\int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx \] Input:
Integrate[(E^((6*x + (-2 - x)*Log[x/(-10 + 8*x^2)])/(12 + 6*x))*(20 - 40*x + 21*x^2 + 64*x^3 + 4*x^4)*Log[3])/(-120*x - 120*x^2 + 66*x^3 + 96*x^4 + 24*x^5),x]
Output:
Integrate[(E^((6*x + (-2 - x)*Log[x/(-10 + 8*x^2)])/(12 + 6*x))*(20 - 40*x + 21*x^2 + 64*x^3 + 4*x^4))/(-120*x - 120*x^2 + 66*x^3 + 96*x^4 + 24*x^5) , x]*Log[3]
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.
| \(\displaystyle \int \frac {\left (4 x^4+64 x^3+21 x^2-40 x+20\right ) \log (3) \exp \left (\frac {(-x-2) \log \left (\frac {x}{8 x^2-10}\right )+6 x}{6 x+12}\right )}{24 x^5+96 x^4+66 x^3-120 x^2-120 x} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \log (3) \int -\frac {e^{\frac {x}{x+2}} \left (-\frac {x}{10-8 x^2}\right )^{-\frac {x+2}{6 x+12}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{6 \left (-4 x^5-16 x^4-11 x^3+20 x^2+20 x\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{6} \log (3) \int \frac {\sqrt [6]{2} e^{\frac {x}{x+2}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{\sqrt [6]{-\frac {x}{5-4 x^2}} \left (-4 x^5-16 x^4-11 x^3+20 x^2+20 x\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\log (3) \int \frac {e^{\frac {x}{x+2}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{\sqrt [6]{-\frac {x}{5-4 x^2}} \left (-4 x^5-16 x^4-11 x^3+20 x^2+20 x\right )}dx}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle -\frac {\log (3) \int \frac {e^{\frac {x}{x+2}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{x \sqrt [6]{-\frac {x}{5-4 x^2}} \left (-4 x^4-16 x^3-11 x^2+20 x+20\right )}dx}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle -\frac {\log (3) \int \left (-\frac {16 e^{\frac {x}{x+2}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{121 x (x+2) \sqrt [6]{-\frac {x}{5-4 x^2}}}-\frac {e^{\frac {x}{x+2}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{11 x (x+2)^2 \sqrt [6]{-\frac {x}{5-4 x^2}}}+\frac {4 e^{\frac {x}{x+2}} (16 x-21) \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{121 x \sqrt [6]{-\frac {x}{5-4 x^2}} \left (4 x^2-5\right )}\right )dx}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\log (3) \int \frac {e^{\frac {x}{x+2}} \left (\frac {x}{4 x^2-5}\right )^{5/6} \left (-4 x^4-64 x^3-21 x^2+40 x-20\right )}{x^2 (x+2)^2}dx}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle -\frac {\left (-\frac {x}{5-4 x^2}\right )^{5/6} \left (4 x^2-5\right )^{5/6} \log (3) \int -\frac {e^{\frac {x}{x+2}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{x^{7/6} (x+2)^2 \left (4 x^2-5\right )^{5/6}}dx}{3\ 2^{5/6} x^{5/6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (-\frac {x}{5-4 x^2}\right )^{5/6} \left (4 x^2-5\right )^{5/6} \log (3) \int \frac {e^{\frac {x}{x+2}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{x^{7/6} (x+2)^2 \left (4 x^2-5\right )^{5/6}}dx}{3\ 2^{5/6} x^{5/6}}\) |
| \(\Big \downarrow \) 7284 |
| \(\displaystyle \frac {\sqrt [6]{2} \left (-\frac {x}{5-4 x^2}\right )^{5/6} \left (4 x^2-5\right )^{5/6} \log (3) \int \frac {e^{\frac {x}{x+2}} \left (4 x^4+64 x^3+21 x^2-40 x+20\right )}{\sqrt [3]{x} (x+2)^2 \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}}{x^{5/6}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt [6]{2} \left (-\frac {x}{5-4 x^2}\right )^{5/6} \left (4 x^2-5\right )^{5/6} \log (3) \int \left (\frac {4 e^{\frac {x}{x+2}} x^{5/3}}{\left (4 x^2-5\right )^{5/6}}+\frac {48 e^{\frac {x}{x+2}} x^{2/3}}{\left (4 x^2-5\right )^{5/6}}-\frac {192 e^{\frac {x}{x+2}} x^{2/3}}{(x+2) \left (4 x^2-5\right )^{5/6}}+\frac {132 e^{\frac {x}{x+2}} x^{2/3}}{(x+2)^2 \left (4 x^2-5\right )^{5/6}}+\frac {5 e^{\frac {x}{x+2}}}{\left (4 x^2-5\right )^{5/6} \sqrt [3]{x}}\right )d\sqrt [6]{x}}{x^{5/6}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [6]{2} \left (-\frac {x}{5-4 x^2}\right )^{5/6} \left (4 x^2-5\right )^{5/6} \log (3) \left (-16 (-2)^{5/6} \int \frac {e^{\frac {x}{x+2}}}{\left (\sqrt [6]{-2}-\sqrt [6]{x}\right ) \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}-16 (-2)^{5/6} \int \frac {e^{\frac {x}{x+2}}}{\left (\sqrt [6]{x}+\sqrt [6]{-2}\right ) \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}+16 i 2^{5/6} \int \frac {e^{\frac {x}{x+2}}}{\left (\sqrt [6]{-2}-\sqrt [3]{-1} \sqrt [6]{x}\right ) \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}+16 i 2^{5/6} \int \frac {e^{\frac {x}{x+2}}}{\left (\sqrt [3]{-1} \sqrt [6]{x}+\sqrt [6]{-2}\right ) \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}-16 \sqrt [6]{-1} 2^{5/6} \int \frac {e^{\frac {x}{x+2}}}{\left (\sqrt [6]{-2}-(-1)^{2/3} \sqrt [6]{x}\right ) \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}-16 \sqrt [6]{-1} 2^{5/6} \int \frac {e^{\frac {x}{x+2}}}{\left ((-1)^{2/3} \sqrt [6]{x}+\sqrt [6]{-2}\right ) \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}+5 \int \frac {e^{\frac {x}{x+2}}}{\sqrt [3]{x} \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}-\frac {33 i \int \frac {e^{\frac {x}{x+2}} \sqrt [6]{x}}{\left (i \sqrt {2}-\sqrt {x}\right )^2 \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}}{\sqrt {2}}+\frac {33 i \int \frac {e^{\frac {x}{x+2}} \sqrt [6]{x}}{\left (\sqrt {x}+i \sqrt {2}\right )^2 \left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}}{\sqrt {2}}+48 \int \frac {e^{\frac {x}{x+2}} x^{2/3}}{\left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}+4 \int \frac {e^{\frac {x}{x+2}} x^{5/3}}{\left (4 x^2-5\right )^{5/6}}d\sqrt [6]{x}\right )}{x^{5/6}}\) |
Input:
Int[(E^((6*x + (-2 - x)*Log[x/(-10 + 8*x^2)])/(12 + 6*x))*(20 - 40*x + 21* x^2 + 64*x^3 + 4*x^4)*Log[3])/(-120*x - 120*x^2 + 66*x^3 + 96*x^4 + 24*x^5 ),x]
Output:
$Aborted
Time = 1.92 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09
| method | result | size |
| default | \(\ln \left (3\right ) {\mathrm e}^{\frac {\left (-2-x \right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}\) | \(35\) |
| parallelrisch | \(\ln \left (3\right ) {\mathrm e}^{\frac {\left (-2-x \right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}\) | \(35\) |
| risch | \(\ln \left (3\right ) {\mathrm e}^{-\frac {\ln \left (\frac {x}{8 x^{2}-10}\right ) x +2 \ln \left (\frac {x}{8 x^{2}-10}\right )-6 x}{6 \left (2+x \right )}}\) | \(44\) |
| gosper | \(\ln \left (3\right ) {\mathrm e}^{-\frac {\ln \left (\frac {x}{8 x^{2}-10}\right ) x +2 \ln \left (\frac {x}{8 x^{2}-10}\right )-6 x}{6 \left (2+x \right )}}\) | \(46\) |
| orering | \(\frac {6 x \left (2+x \right )^{2} \left (4 x^{2}-5\right ) \ln \left (3\right ) {\mathrm e}^{\frac {\left (-2-x \right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}}{24 x^{5}+96 x^{4}+66 x^{3}-120 x^{2}-120 x}\) | \(75\) |
| norman | \(\frac {x \ln \left (3\right ) {\mathrm e}^{\frac {\left (-2-x \right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}+2 \ln \left (3\right ) {\mathrm e}^{\frac {\left (-2-x \right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}}{2+x}\) | \(78\) |
Input:
int((4*x^4+64*x^3+21*x^2-40*x+20)*ln(3)*exp(((-2-x)*ln(x/(8*x^2-10))+6*x)/ (6*x+12))/(24*x^5+96*x^4+66*x^3-120*x^2-120*x),x,method=_RETURNVERBOSE)
Output:
ln(3)*exp(1/6/(2+x)*((-2-x)*ln(1/2*x/(4*x^2-5))+6*x))
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx=e^{\left (-\frac {{\left (x + 2\right )} \log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right ) - 6 \, x}{6 \, {\left (x + 2\right )}}\right )} \log \left (3\right ) \] Input:
integrate((4*x^4+64*x^3+21*x^2-40*x+20)*log(3)*exp(((-2-x)*log(x/(8*x^2-10 ))+6*x)/(6*x+12))/(24*x^5+96*x^4+66*x^3-120*x^2-120*x),x, algorithm="frica s")
Output:
e^(-1/6*((x + 2)*log(1/2*x/(4*x^2 - 5)) - 6*x)/(x + 2))*log(3)
Time = 0.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx=e^{\frac {6 x + \left (- x - 2\right ) \log {\left (\frac {x}{8 x^{2} - 10} \right )}}{6 x + 12}} \log {\left (3 \right )} \] Input:
integrate((4*x**4+64*x**3+21*x**2-40*x+20)*ln(3)*exp(((-2-x)*ln(x/(8*x**2- 10))+6*x)/(6*x+12))/(24*x**5+96*x**4+66*x**3-120*x**2-120*x),x)
Output:
exp((6*x + (-x - 2)*log(x/(8*x**2 - 10)))/(6*x + 12))*log(3)
\[ \int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx=\int { \frac {{\left (4 \, x^{4} + 64 \, x^{3} + 21 \, x^{2} - 40 \, x + 20\right )} e^{\left (-\frac {{\left (x + 2\right )} \log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right ) - 6 \, x}{6 \, {\left (x + 2\right )}}\right )} \log \left (3\right )}{6 \, {\left (4 \, x^{5} + 16 \, x^{4} + 11 \, x^{3} - 20 \, x^{2} - 20 \, x\right )}} \,d x } \] Input:
integrate((4*x^4+64*x^3+21*x^2-40*x+20)*log(3)*exp(((-2-x)*log(x/(8*x^2-10 ))+6*x)/(6*x+12))/(24*x^5+96*x^4+66*x^3-120*x^2-120*x),x, algorithm="maxim a")
Output:
1/6*integrate((4*x^4 + 64*x^3 + 21*x^2 - 40*x + 20)*e^(-1/6*((x + 2)*log(1 /2*x/(4*x^2 - 5)) - 6*x)/(x + 2))/(4*x^5 + 16*x^4 + 11*x^3 - 20*x^2 - 20*x ), x)*log(3)
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.66 \[ \int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx=e^{\left (-\frac {x \log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right )}{6 \, {\left (x + 2\right )}} + \frac {x}{x + 2} - \frac {\log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right )}{3 \, {\left (x + 2\right )}}\right )} \log \left (3\right ) \] Input:
integrate((4*x^4+64*x^3+21*x^2-40*x+20)*log(3)*exp(((-2-x)*log(x/(8*x^2-10 ))+6*x)/(6*x+12))/(24*x^5+96*x^4+66*x^3-120*x^2-120*x),x, algorithm="giac" )
Output:
e^(-1/6*x*log(1/2*x/(4*x^2 - 5))/(x + 2) + x/(x + 2) - 1/3*log(1/2*x/(4*x^ 2 - 5))/(x + 2))*log(3)
Timed out. \[ \int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx=\int \frac {{\mathrm {e}}^{\frac {6\,x-\ln \left (\frac {x}{8\,x^2-10}\right )\,\left (x+2\right )}{6\,x+12}}\,\ln \left (3\right )\,\left (4\,x^4+64\,x^3+21\,x^2-40\,x+20\right )}{24\,x^5+96\,x^4+66\,x^3-120\,x^2-120\,x} \,d x \] Input:
int((exp((6*x - log(x/(8*x^2 - 10))*(x + 2))/(6*x + 12))*log(3)*(21*x^2 - 40*x + 64*x^3 + 4*x^4 + 20))/(66*x^3 - 120*x^2 - 120*x + 96*x^4 + 24*x^5), x)
Output:
int((exp((6*x - log(x/(8*x^2 - 10))*(x + 2))/(6*x + 12))*log(3)*(21*x^2 - 40*x + 64*x^3 + 4*x^4 + 20))/(66*x^3 - 120*x^2 - 120*x + 96*x^4 + 24*x^5), x)
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx=\frac {\mathrm {log}\left (3\right ) e}{e^{\frac {\mathrm {log}\left (\frac {x}{8 x^{2}-10}\right ) x +2 \,\mathrm {log}\left (\frac {x}{8 x^{2}-10}\right )+12}{6 x +12}}} \] Input:
int((4*x^4+64*x^3+21*x^2-40*x+20)*log(3)*exp(((-2-x)*log(x/(8*x^2-10))+6*x )/(6*x+12))/(24*x^5+96*x^4+66*x^3-120*x^2-120*x),x)
Output:
(log(3)*e)/e**((log(x/(8*x**2 - 10))*x + 2*log(x/(8*x**2 - 10)) + 12)/(6*x + 12))