Integrand size = 81, antiderivative size = 31 \begin {dmath*} \int \frac {-4+4 x^2+e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} \left (6 x+3 e^{e^x+x} x\right )}{2 e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} x+2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )} \, dx=\log \left (e^{3 \left (-2+\frac {e^{e^x}}{2}+x\right )}+x^2-\log \left (\frac {4 x^2}{25}\right )\right ) \end {dmath*}
Time = 0.84 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \begin {dmath*} \int \frac {-4+4 x^2+e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} \left (6 x+3 e^{e^x+x} x\right )}{2 e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} x+2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )} \, dx=\log \left (e^{\frac {3 e^{e^x}}{2}+3 x}+e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )\right ) \end {dmath*}
Integrate[(-4 + 4*x^2 + E^((-12 + 3*E^E^x + 6*x)/2)*(6*x + 3*E^(E^x + x)*x ))/(2*E^((-12 + 3*E^E^x + 6*x)/2)*x + 2*x^3 - 2*x*Log[(4*x^2)/25]),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.
\(\displaystyle \int \frac {4 x^2+e^{\frac {1}{2} \left (6 x+3 e^{e^x}-12\right )} \left (3 e^{x+e^x} x+6 x\right )-4}{2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )+2 e^{\frac {1}{2} \left (6 x+3 e^{e^x}-12\right )} x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {e^6 \left (3 e^{x+e^x} x^3+6 x^3-4 x^2-3 e^{x+e^x} x \log \left (\frac {4 x^2}{25}\right )-6 x \log \left (\frac {4 x^2}{25}\right )+4\right )}{2 x \left (e^6 x^2-e^6 \log \left (\frac {4 x^2}{25}\right )+e^{3 x+\frac {3 e^{e^x}}{2}}\right )}+\frac {3 e^{x+e^x}}{2}+3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 e^6 \int \frac {1}{x \left (e^6 x^2+e^{3 x+\frac {3 e^{e^x}}{2}}-e^6 \log \left (\frac {4 x^2}{25}\right )\right )}dx+2 e^6 \int \frac {x}{e^6 x^2+e^{3 x+\frac {3 e^{e^x}}{2}}-e^6 \log \left (\frac {4 x^2}{25}\right )}dx-3 e^6 \int \frac {x^2}{e^6 x^2+e^{3 x+\frac {3 e^{e^x}}{2}}-e^6 \log \left (\frac {4 x^2}{25}\right )}dx-\frac {3}{2} e^6 \int \frac {e^{x+e^x} x^2}{e^6 x^2+e^{3 x+\frac {3 e^{e^x}}{2}}-e^6 \log \left (\frac {4 x^2}{25}\right )}dx+\frac {3}{2} e^6 \int \frac {e^{x+e^x} \log \left (\frac {4 x^2}{25}\right )}{e^6 x^2+e^{3 x+\frac {3 e^{e^x}}{2}}-e^6 \log \left (\frac {4 x^2}{25}\right )}dx-3 e^6 \int \frac {\log \left (\frac {4 x^2}{25}\right )}{-e^6 x^2-e^{3 x+\frac {3 e^{e^x}}{2}}+e^6 \log \left (\frac {4 x^2}{25}\right )}dx+3 x+\frac {3 e^{e^x}}{2}\) |
Int[(-4 + 4*x^2 + E^((-12 + 3*E^E^x + 6*x)/2)*(6*x + 3*E^(E^x + x)*x))/(2* E^((-12 + 3*E^E^x + 6*x)/2)*x + 2*x^3 - 2*x*Log[(4*x^2)/25]),x]
3.7.15.3.1 Defintions of rubi rules used
Time = 2.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(\ln \left (x^{2}+{\mathrm e}^{\frac {3 \,{\mathrm e}^{{\mathrm e}^{x}}}{2}+3 x -6}-\ln \left (\frac {4 x^{2}}{25}\right )\right )\) | \(25\) |
risch | \(6+\ln \left ({\mathrm e}^{\frac {3 \,{\mathrm e}^{{\mathrm e}^{x}}}{2}+3 x -6}-\frac {i \left (-\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+2 i x^{2}-4 i \ln \left (2\right )+4 i \ln \left (5\right )-4 i \ln \left (x \right )\right )}{2}\right )\) | \(87\) |
int(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x^2-4)/(2*x *exp(3/2*exp(exp(x))+3*x-6)-2*x*ln(4/25*x^2)+2*x^3),x,method=_RETURNVERBOS E)
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \begin {dmath*} \int \frac {-4+4 x^2+e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} \left (6 x+3 e^{e^x+x} x\right )}{2 e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} x+2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )} \, dx=\log \left (x^{2} + e^{\left (\frac {3}{2} \, {\left (2 \, {\left (x - 2\right )} e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} - \log \left (\frac {4}{25} \, x^{2}\right )\right ) \end {dmath*}
integrate(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x^2-4 )/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2*x*log(4/25*x^2)+2*x^3),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \begin {dmath*} \int \frac {-4+4 x^2+e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} \left (6 x+3 e^{e^x+x} x\right )}{2 e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} x+2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )} \, dx=\log {\left (x^{2} + e^{3 x + \frac {3 e^{e^{x}}}{2} - 6} - \log {\left (\frac {4 x^{2}}{25} \right )} \right )} \end {dmath*}
integrate(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x**2- 4)/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2*x*ln(4/25*x**2)+2*x**3),x)
Time = 0.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \begin {dmath*} \int \frac {-4+4 x^2+e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} \left (6 x+3 e^{e^x+x} x\right )}{2 e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} x+2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )} \, dx=3 \, x + \log \left ({\left (x^{2} e^{6} + 2 \, {\left (\log \left (5\right ) - \log \left (2\right )\right )} e^{6} - 2 \, e^{6} \log \left (x\right ) + e^{\left (3 \, x + \frac {3}{2} \, e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-3 \, x\right )}\right ) \end {dmath*}
integrate(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x^2-4 )/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2*x*log(4/25*x^2)+2*x^3),x, algorithm=\
3*x + log((x^2*e^6 + 2*(log(5) - log(2))*e^6 - 2*e^6*log(x) + e^(3*x + 3/2 *e^(e^x)))*e^(-3*x))
\begin {dmath*} \int \frac {-4+4 x^2+e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} \left (6 x+3 e^{e^x+x} x\right )}{2 e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} x+2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )} \, dx=\int { \frac {4 \, x^{2} + 3 \, {\left (x e^{\left (x + e^{x}\right )} + 2 \, x\right )} e^{\left (3 \, x + \frac {3}{2} \, e^{\left (e^{x}\right )} - 6\right )} - 4}{2 \, {\left (x^{3} + x e^{\left (3 \, x + \frac {3}{2} \, e^{\left (e^{x}\right )} - 6\right )} - x \log \left (\frac {4}{25} \, x^{2}\right )\right )}} \,d x } \end {dmath*}
integrate(((3*x*exp(x)*exp(exp(x))+6*x)*exp(3/2*exp(exp(x))+3*x-6)+4*x^2-4 )/(2*x*exp(3/2*exp(exp(x))+3*x-6)-2*x*log(4/25*x^2)+2*x^3),x, algorithm=\
integrate(1/2*(4*x^2 + 3*(x*e^(x + e^x) + 2*x)*e^(3*x + 3/2*e^(e^x) - 6) - 4)/(x^3 + x*e^(3*x + 3/2*e^(e^x) - 6) - x*log(4/25*x^2)), x)
Time = 15.75 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \begin {dmath*} \int \frac {-4+4 x^2+e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} \left (6 x+3 e^{e^x+x} x\right )}{2 e^{\frac {1}{2} \left (-12+3 e^{e^x}+6 x\right )} x+2 x^3-2 x \log \left (\frac {4 x^2}{25}\right )} \, dx=\ln \left (x^2-\ln \left (\frac {4\,x^2}{25}\right )+{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-6}\,{\left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}\right )}^{3/2}\right ) \end {dmath*}