Integrand size = 103, antiderivative size = 31 \[ \int \frac {-384+(-384+3 x) \log \left (\frac {16 x}{-128+x}\right )+e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{-128+x}\right )}{-1152+9 x+e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{-128+x}\right )+e^{32/x} \left (-128 x^2+x^3\right ) \log ^2\left (\frac {16 x}{-128+x}\right )} \, dx=\frac {x}{-e^{16/x} x+\frac {3}{\log \left (\frac {x}{-8+\frac {x}{16}}\right )}} \]
Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-384+(-384+3 x) \log \left (\frac {16 x}{-128+x}\right )+e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{-128+x}\right )}{-1152+9 x+e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{-128+x}\right )+e^{32/x} \left (-128 x^2+x^3\right ) \log ^2\left (\frac {16 x}{-128+x}\right )} \, dx=-\frac {x \log \left (\frac {16 x}{-128+x}\right )}{-3+e^{16/x} x \log \left (\frac {16 x}{-128+x}\right )} \]
Integrate[(-384 + (-384 + 3*x)*Log[(16*x)/(-128 + x)] + E^(16/x)*(2048 - 1 6*x)*Log[(16*x)/(-128 + x)]^2)/(-1152 + 9*x + E^(16/x)*(768*x - 6*x^2)*Log [(16*x)/(-128 + x)] + E^(32/x)*(-128*x^2 + x^3)*Log[(16*x)/(-128 + x)]^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.
\(\displaystyle \int \frac {e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{x-128}\right )+(3 x-384) \log \left (\frac {16 x}{x-128}\right )-384}{e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{x-128}\right )+e^{32/x} \left (x^3-128 x^2\right ) \log ^2\left (\frac {16 x}{x-128}\right )+9 x-1152} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{x-128}\right )-(3 x-384) \log \left (\frac {16 x}{x-128}\right )+384}{(128-x) \left (3-e^{16/x} x \log \left (\frac {16 x}{x-128}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 \left (x^2 \log \left (\frac {16 x}{x-128}\right )-128 x-144 x \log \left (\frac {16 x}{x-128}\right )+2048 \log \left (\frac {16 x}{x-128}\right )\right )}{(x-128) x \left (e^{16/x} x \log \left (\frac {16 x}{x-128}\right )-3\right )^2}-\frac {16 \log \left (\frac {16 x}{x-128}\right )}{x \left (e^{16/x} x \log \left (\frac {16 x}{x-128}\right )-3\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -384 \int \frac {1}{(x-128) \left (e^{16/x} x \log \left (\frac {16 x}{x-128}\right )-3\right )^2}dx+3 \int \frac {\log \left (\frac {16 x}{x-128}\right )}{\left (e^{16/x} x \log \left (\frac {16 x}{x-128}\right )-3\right )^2}dx-48 \int \frac {\log \left (\frac {16 x}{x-128}\right )}{x \left (e^{16/x} x \log \left (\frac {16 x}{x-128}\right )-3\right )^2}dx-16 \int \frac {\log \left (\frac {16 x}{x-128}\right )}{x \left (e^{16/x} x \log \left (\frac {16 x}{x-128}\right )-3\right )}dx\) |
Int[(-384 + (-384 + 3*x)*Log[(16*x)/(-128 + x)] + E^(16/x)*(2048 - 16*x)*L og[(16*x)/(-128 + x)]^2)/(-1152 + 9*x + E^(16/x)*(768*x - 6*x^2)*Log[(16*x )/(-128 + x)] + E^(32/x)*(-128*x^2 + x^3)*Log[(16*x)/(-128 + x)]^2),x]
3.23.75.3.1 Defintions of rubi rules used
Time = 0.81 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10
method | result | size |
parallelrisch | \(-\frac {\ln \left (\frac {16 x}{x -128}\right ) x}{{\mathrm e}^{\frac {16}{x}} \ln \left (\frac {16 x}{x -128}\right ) x -3}\) | \(34\) |
risch | \(-{\mathrm e}^{-\frac {16}{x}}+\frac {6 i {\mathrm e}^{-\frac {16}{x}}}{\pi x \,\operatorname {csgn}\left (\frac {i}{x -128}\right ) \operatorname {csgn}\left (\frac {i x}{x -128}\right )^{2} {\mathrm e}^{\frac {16}{x}}-\pi x \,\operatorname {csgn}\left (\frac {i}{x -128}\right ) \operatorname {csgn}\left (\frac {i x}{x -128}\right ) \operatorname {csgn}\left (i x \right ) {\mathrm e}^{\frac {16}{x}}-\pi x \operatorname {csgn}\left (\frac {i x}{x -128}\right )^{3} {\mathrm e}^{\frac {16}{x}}+\pi x \operatorname {csgn}\left (\frac {i x}{x -128}\right )^{2} \operatorname {csgn}\left (i x \right ) {\mathrm e}^{\frac {16}{x}}-8 i \ln \left (2\right ) x \,{\mathrm e}^{\frac {16}{x}}-2 i \ln \left (x \right ) x \,{\mathrm e}^{\frac {16}{x}}+2 i x \,{\mathrm e}^{\frac {16}{x}} \ln \left (x -128\right )+6 i}\) | \(174\) |
int(((-16*x+2048)*exp(16/x)*ln(16*x/(x-128))^2+(3*x-384)*ln(16*x/(x-128))- 384)/((x^3-128*x^2)*exp(16/x)^2*ln(16*x/(x-128))^2+(-6*x^2+768*x)*exp(16/x )*ln(16*x/(x-128))+9*x-1152),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-384+(-384+3 x) \log \left (\frac {16 x}{-128+x}\right )+e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{-128+x}\right )}{-1152+9 x+e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{-128+x}\right )+e^{32/x} \left (-128 x^2+x^3\right ) \log ^2\left (\frac {16 x}{-128+x}\right )} \, dx=-\frac {x \log \left (\frac {16 \, x}{x - 128}\right )}{x e^{\frac {16}{x}} \log \left (\frac {16 \, x}{x - 128}\right ) - 3} \]
integrate(((-16*x+2048)*exp(16/x)*log(16*x/(x-128))^2+(3*x-384)*log(16*x/( x-128))-384)/((x^3-128*x^2)*exp(16/x)^2*log(16*x/(x-128))^2+(-6*x^2+768*x) *exp(16/x)*log(16*x/(x-128))+9*x-1152),x, algorithm=\
Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-384+(-384+3 x) \log \left (\frac {16 x}{-128+x}\right )+e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{-128+x}\right )}{-1152+9 x+e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{-128+x}\right )+e^{32/x} \left (-128 x^2+x^3\right ) \log ^2\left (\frac {16 x}{-128+x}\right )} \, dx=- \frac {x \log {\left (\frac {16 x}{x - 128} \right )}}{x e^{\frac {16}{x}} \log {\left (\frac {16 x}{x - 128} \right )} - 3} \]
integrate(((-16*x+2048)*exp(16/x)*ln(16*x/(x-128))**2+(3*x-384)*ln(16*x/(x -128))-384)/((x**3-128*x**2)*exp(16/x)**2*ln(16*x/(x-128))**2+(-6*x**2+768 *x)*exp(16/x)*ln(16*x/(x-128))+9*x-1152),x)
Time = 0.34 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-384+(-384+3 x) \log \left (\frac {16 x}{-128+x}\right )+e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{-128+x}\right )}{-1152+9 x+e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{-128+x}\right )+e^{32/x} \left (-128 x^2+x^3\right ) \log ^2\left (\frac {16 x}{-128+x}\right )} \, dx=-\frac {4 \, x \log \left (2\right ) - x \log \left (x - 128\right ) + x \log \left (x\right )}{{\left (4 \, x \log \left (2\right ) - x \log \left (x - 128\right ) + x \log \left (x\right )\right )} e^{\frac {16}{x}} - 3} \]
integrate(((-16*x+2048)*exp(16/x)*log(16*x/(x-128))^2+(3*x-384)*log(16*x/( x-128))-384)/((x^3-128*x^2)*exp(16/x)^2*log(16*x/(x-128))^2+(-6*x^2+768*x) *exp(16/x)*log(16*x/(x-128))+9*x-1152),x, algorithm=\
-(4*x*log(2) - x*log(x - 128) + x*log(x))/((4*x*log(2) - x*log(x - 128) + x*log(x))*e^(16/x) - 3)
Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {-384+(-384+3 x) \log \left (\frac {16 x}{-128+x}\right )+e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{-128+x}\right )}{-1152+9 x+e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{-128+x}\right )+e^{32/x} \left (-128 x^2+x^3\right ) \log ^2\left (\frac {16 x}{-128+x}\right )} \, dx=-\frac {3}{x e^{\frac {32}{x}} \log \left (\frac {16 \, x}{x - 128}\right ) - 3 \, e^{\frac {16}{x}}} - e^{\left (-\frac {16}{x}\right )} \]
integrate(((-16*x+2048)*exp(16/x)*log(16*x/(x-128))^2+(3*x-384)*log(16*x/( x-128))-384)/((x^3-128*x^2)*exp(16/x)^2*log(16*x/(x-128))^2+(-6*x^2+768*x) *exp(16/x)*log(16*x/(x-128))+9*x-1152),x, algorithm=\
Time = 8.78 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-384+(-384+3 x) \log \left (\frac {16 x}{-128+x}\right )+e^{16/x} (2048-16 x) \log ^2\left (\frac {16 x}{-128+x}\right )}{-1152+9 x+e^{16/x} \left (768 x-6 x^2\right ) \log \left (\frac {16 x}{-128+x}\right )+e^{32/x} \left (-128 x^2+x^3\right ) \log ^2\left (\frac {16 x}{-128+x}\right )} \, dx=-\frac {x\,\ln \left (\frac {16\,x}{x-128}\right )}{x\,{\mathrm {e}}^{16/x}\,\ln \left (\frac {16\,x}{x-128}\right )-3} \]