Integrand size = 86, antiderivative size = 28 \begin {dmath*} \int \frac {-11552 x^3-4864 x^3 \log \left (x^2\right )-512 x^3 \log ^2\left (x^2\right )+e^{-\frac {1}{19+4 \log \left (x^2\right )}} \left (-353-152 \log \left (x^2\right )-16 \log ^2\left (x^2\right )\right )}{361 x^2+152 x^2 \log \left (x^2\right )+16 x^2 \log ^2\left (x^2\right )} \, dx=2+\frac {e^{\frac {x}{x-4 x \left (5+\log \left (x^2\right )\right )}}}{x}-16 x^2 \end {dmath*}
Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \begin {dmath*} \int \frac {-11552 x^3-4864 x^3 \log \left (x^2\right )-512 x^3 \log ^2\left (x^2\right )+e^{-\frac {1}{19+4 \log \left (x^2\right )}} \left (-353-152 \log \left (x^2\right )-16 \log ^2\left (x^2\right )\right )}{361 x^2+152 x^2 \log \left (x^2\right )+16 x^2 \log ^2\left (x^2\right )} \, dx=\frac {e^{-\frac {1}{19+4 \log \left (x^2\right )}}}{x}-16 x^2 \end {dmath*}
Integrate[(-11552*x^3 - 4864*x^3*Log[x^2] - 512*x^3*Log[x^2]^2 + (-353 - 1 52*Log[x^2] - 16*Log[x^2]^2)/E^(19 + 4*Log[x^2])^(-1))/(361*x^2 + 152*x^2* Log[x^2] + 16*x^2*Log[x^2]^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.
\(\displaystyle \int \frac {-11552 x^3+e^{-\frac {1}{4 \log \left (x^2\right )+19}} \left (-16 \log ^2\left (x^2\right )-152 \log \left (x^2\right )-353\right )-512 x^3 \log ^2\left (x^2\right )-4864 x^3 \log \left (x^2\right )}{361 x^2+16 x^2 \log ^2\left (x^2\right )+152 x^2 \log \left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-11552 x^3+e^{-\frac {1}{4 \log \left (x^2\right )+19}} \left (-16 \log ^2\left (x^2\right )-152 \log \left (x^2\right )-353\right )-512 x^3 \log ^2\left (x^2\right )-4864 x^3 \log \left (x^2\right )}{x^2 \left (4 \log \left (x^2\right )+19\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {e^{\frac {1}{-4 \log \left (x^2\right )-19}} \left (16 \log ^2\left (x^2\right )+152 \log \left (x^2\right )+353\right )}{x^2 \left (4 \log \left (x^2\right )+19\right )^2}-32 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {e^{\frac {1}{-4 \log \left (x^2\right )-19}}}{x^2}dx+8 \int \frac {e^{\frac {1}{-4 \log \left (x^2\right )-19}}}{x^2 \left (4 \log \left (x^2\right )+19\right )^2}dx-16 x^2\) |
Int[(-11552*x^3 - 4864*x^3*Log[x^2] - 512*x^3*Log[x^2]^2 + (-353 - 152*Log [x^2] - 16*Log[x^2]^2)/E^(19 + 4*Log[x^2])^(-1))/(361*x^2 + 152*x^2*Log[x^ 2] + 16*x^2*Log[x^2]^2),x]
3.7.83.3.1 Defintions of rubi rules used
Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-16 x^{2}+\frac {{\mathrm e}^{-\frac {1}{4 \ln \left (x^{2}\right )+19}}}{x}\) | \(24\) |
parallelrisch | \(\frac {-77824 x^{3}+4864 \,{\mathrm e}^{-\frac {1}{4 \ln \left (x^{2}\right )+19}}}{4864 x}\) | \(27\) |
default | \(-16 x^{2}+\frac {\left (4 \ln \left (x^{2}\right )-8 \ln \left (x \right )+19\right ) {\mathrm e}^{-\frac {1}{4 \ln \left (x^{2}\right )+19}}+8 \ln \left (x \right ) {\mathrm e}^{-\frac {1}{4 \ln \left (x^{2}\right )+19}}}{x \left (4 \ln \left (x^{2}\right )+19\right )}\) | \(65\) |
parts | \(-16 x^{2}+\frac {\left (4 \ln \left (x^{2}\right )-8 \ln \left (x \right )+19\right ) {\mathrm e}^{-\frac {1}{4 \ln \left (x^{2}\right )+19}}+8 \ln \left (x \right ) {\mathrm e}^{-\frac {1}{4 \ln \left (x^{2}\right )+19}}}{x \left (4 \ln \left (x^{2}\right )+19\right )}\) | \(65\) |
int(((-16*ln(x^2)^2-152*ln(x^2)-353)*exp(-1/(4*ln(x^2)+19))-512*x^3*ln(x^2 )^2-4864*x^3*ln(x^2)-11552*x^3)/(16*x^2*ln(x^2)^2+152*x^2*ln(x^2)+361*x^2) ,x,method=_RETURNVERBOSE)
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \begin {dmath*} \int \frac {-11552 x^3-4864 x^3 \log \left (x^2\right )-512 x^3 \log ^2\left (x^2\right )+e^{-\frac {1}{19+4 \log \left (x^2\right )}} \left (-353-152 \log \left (x^2\right )-16 \log ^2\left (x^2\right )\right )}{361 x^2+152 x^2 \log \left (x^2\right )+16 x^2 \log ^2\left (x^2\right )} \, dx=-\frac {16 \, x^{3} - e^{\left (-\frac {1}{4 \, \log \left (x^{2}\right ) + 19}\right )}}{x} \end {dmath*}
integrate(((-16*log(x^2)^2-152*log(x^2)-353)*exp(-1/(4*log(x^2)+19))-512*x ^3*log(x^2)^2-4864*x^3*log(x^2)-11552*x^3)/(16*x^2*log(x^2)^2+152*x^2*log( x^2)+361*x^2),x, algorithm=\
Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \begin {dmath*} \int \frac {-11552 x^3-4864 x^3 \log \left (x^2\right )-512 x^3 \log ^2\left (x^2\right )+e^{-\frac {1}{19+4 \log \left (x^2\right )}} \left (-353-152 \log \left (x^2\right )-16 \log ^2\left (x^2\right )\right )}{361 x^2+152 x^2 \log \left (x^2\right )+16 x^2 \log ^2\left (x^2\right )} \, dx=- 16 x^{2} + \frac {e^{- \frac {1}{4 \log {\left (x^{2} \right )} + 19}}}{x} \end {dmath*}
integrate(((-16*ln(x**2)**2-152*ln(x**2)-353)*exp(-1/(4*ln(x**2)+19))-512* x**3*ln(x**2)**2-4864*x**3*ln(x**2)-11552*x**3)/(16*x**2*ln(x**2)**2+152*x **2*ln(x**2)+361*x**2),x)
\begin {dmath*} \int \frac {-11552 x^3-4864 x^3 \log \left (x^2\right )-512 x^3 \log ^2\left (x^2\right )+e^{-\frac {1}{19+4 \log \left (x^2\right )}} \left (-353-152 \log \left (x^2\right )-16 \log ^2\left (x^2\right )\right )}{361 x^2+152 x^2 \log \left (x^2\right )+16 x^2 \log ^2\left (x^2\right )} \, dx=\int { -\frac {512 \, x^{3} \log \left (x^{2}\right )^{2} + 4864 \, x^{3} \log \left (x^{2}\right ) + 11552 \, x^{3} + {\left (16 \, \log \left (x^{2}\right )^{2} + 152 \, \log \left (x^{2}\right ) + 353\right )} e^{\left (-\frac {1}{4 \, \log \left (x^{2}\right ) + 19}\right )}}{16 \, x^{2} \log \left (x^{2}\right )^{2} + 152 \, x^{2} \log \left (x^{2}\right ) + 361 \, x^{2}} \,d x } \end {dmath*}
integrate(((-16*log(x^2)^2-152*log(x^2)-353)*exp(-1/(4*log(x^2)+19))-512*x ^3*log(x^2)^2-4864*x^3*log(x^2)-11552*x^3)/(16*x^2*log(x^2)^2+152*x^2*log( x^2)+361*x^2),x, algorithm=\
-16*x^2 - integrate((64*log(x)^2 + 304*log(x) + 353)*e^(-1/(8*log(x) + 19) )/(64*x^2*log(x)^2 + 304*x^2*log(x) + 361*x^2), x)
\begin {dmath*} \int \frac {-11552 x^3-4864 x^3 \log \left (x^2\right )-512 x^3 \log ^2\left (x^2\right )+e^{-\frac {1}{19+4 \log \left (x^2\right )}} \left (-353-152 \log \left (x^2\right )-16 \log ^2\left (x^2\right )\right )}{361 x^2+152 x^2 \log \left (x^2\right )+16 x^2 \log ^2\left (x^2\right )} \, dx=\int { -\frac {512 \, x^{3} \log \left (x^{2}\right )^{2} + 4864 \, x^{3} \log \left (x^{2}\right ) + 11552 \, x^{3} + {\left (16 \, \log \left (x^{2}\right )^{2} + 152 \, \log \left (x^{2}\right ) + 353\right )} e^{\left (-\frac {1}{4 \, \log \left (x^{2}\right ) + 19}\right )}}{16 \, x^{2} \log \left (x^{2}\right )^{2} + 152 \, x^{2} \log \left (x^{2}\right ) + 361 \, x^{2}} \,d x } \end {dmath*}
integrate(((-16*log(x^2)^2-152*log(x^2)-353)*exp(-1/(4*log(x^2)+19))-512*x ^3*log(x^2)^2-4864*x^3*log(x^2)-11552*x^3)/(16*x^2*log(x^2)^2+152*x^2*log( x^2)+361*x^2),x, algorithm=\
integrate(-(512*x^3*log(x^2)^2 + 4864*x^3*log(x^2) + 11552*x^3 + (16*log(x ^2)^2 + 152*log(x^2) + 353)*e^(-1/(4*log(x^2) + 19)))/(16*x^2*log(x^2)^2 + 152*x^2*log(x^2) + 361*x^2), x)
Time = 15.97 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \begin {dmath*} \int \frac {-11552 x^3-4864 x^3 \log \left (x^2\right )-512 x^3 \log ^2\left (x^2\right )+e^{-\frac {1}{19+4 \log \left (x^2\right )}} \left (-353-152 \log \left (x^2\right )-16 \log ^2\left (x^2\right )\right )}{361 x^2+152 x^2 \log \left (x^2\right )+16 x^2 \log ^2\left (x^2\right )} \, dx=\frac {{\mathrm {e}}^{-\frac {1}{\ln \left (x^8\right )+19}}}{x}-16\,x^2 \end {dmath*}