Integrand size = 121, antiderivative size = 32 \begin {dmath*} \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 e^{x-\frac {-\frac {x^3}{3}+\frac {x}{3-\log \left (x^2\right )}}{e}} x \end {dmath*}
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \begin {dmath*} \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 e^{\frac {x \left (-3 \left (-1+3 e+x^2\right )+\left (3 e+x^2\right ) \log \left (x^2\right )\right )}{3 e \left (-3+\log \left (x^2\right )\right )}} x \end {dmath*}
Integrate[(E^((3*x - 9*E*x - 3*x^3 + (3*E*x + x^3)*Log[x^2])/(-9*E + 3*E*L og[x^2]))*(-20*x + 36*x^3 + E*(36 + 36*x) + (E*(-24 - 24*x) + 4*x - 24*x^3 )*Log[x^2] + (4*x^3 + E*(4 + 4*x))*Log[x^2]^2))/(9*E - 6*E*Log[x^2] + E*Lo g[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.
\(\displaystyle \int \frac {\left (36 x^3+\left (4 x^3+e (4 x+4)\right ) \log ^2\left (x^2\right )+\left (-24 x^3+4 x+e (-24 x-24)\right ) \log \left (x^2\right )-20 x+e (36 x+36)\right ) \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )-9 e x+3 x}{3 e \log \left (x^2\right )-9 e}\right )}{e \log ^2\left (x^2\right )-6 e \log \left (x^2\right )+9 e} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (36 x^3+\left (4 x^3+e (4 x+4)\right ) \log ^2\left (x^2\right )+\left (-24 x^3+4 x+e (-24 x-24)\right ) \log \left (x^2\right )-20 x+e (36 x+36)\right ) \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )+3 (1-3 e) x}{3 e \left (\log \left (x^2\right )-3\right )}-1\right )}{\left (3-\log \left (x^2\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 x \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )+3 (1-3 e) x}{3 e \left (\log \left (x^2\right )-3\right )}-1\right )}{\log \left (x^2\right )-3}-\frac {8 x \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )+3 (1-3 e) x}{3 e \left (\log \left (x^2\right )-3\right )}-1\right )}{\left (\log \left (x^2\right )-3\right )^2}+4 \left (x^3+e x+e\right ) \exp \left (\frac {-3 x^3+\left (x^3+3 e x\right ) \log \left (x^2\right )+3 (1-3 e) x}{3 e \left (\log \left (x^2\right )-3\right )}-1\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}\right )dx+4 \int \exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}\right ) xdx+4 \int \exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}-1\right ) x^3dx-8 \int \frac {\exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}-1\right ) x}{\left (\log \left (x^2\right )-3\right )^2}dx+4 \int \frac {\exp \left (\frac {-3 x^3+3 (1-3 e) x+\left (x^3+3 e x\right ) \log \left (x^2\right )}{3 e \left (\log \left (x^2\right )-3\right )}-1\right ) x}{\log \left (x^2\right )-3}dx\) |
Int[(E^((3*x - 9*E*x - 3*x^3 + (3*E*x + x^3)*Log[x^2])/(-9*E + 3*E*Log[x^2 ]))*(-20*x + 36*x^3 + E*(36 + 36*x) + (E*(-24 - 24*x) + 4*x - 24*x^3)*Log[ x^2] + (4*x^3 + E*(4 + 4*x))*Log[x^2]^2))/(9*E - 6*E*Log[x^2] + E*Log[x^2] ^2),x]
3.12.7.3.1 Defintions of rubi rules used
Time = 3.88 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41
method | result | size |
risch | \(4 x \,{\mathrm e}^{\frac {x \left (x^{2} \ln \left (x^{2}\right )+3 \,{\mathrm e} \ln \left (x^{2}\right )-3 x^{2}-9 \,{\mathrm e}+3\right ) {\mathrm e}^{-1}}{3 \ln \left (x^{2}\right )-9}}\) | \(45\) |
parallelrisch | \(4 \,{\mathrm e}^{\frac {\left (\left (3 x \,{\mathrm e}+x^{3}\right ) \ln \left (x^{2}\right )-9 x \,{\mathrm e}-3 x^{3}+3 x \right ) {\mathrm e}^{-1}}{3 \ln \left (x^{2}\right )-9}} x\) | \(47\) |
int((((4+4*x)*exp(1)+4*x^3)*ln(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x)*ln(x^ 2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*x*exp(1)+x^3)*ln(x^2)-9*x*exp(1)- 3*x^3+3*x)/(3*exp(1)*ln(x^2)-9*exp(1)))/(exp(1)*ln(x^2)^2-6*exp(1)*ln(x^2) +9*exp(1)),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \begin {dmath*} \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 \, x e^{\left (-\frac {3 \, x^{3} + 9 \, x e - {\left (x^{3} + 3 \, x e\right )} \log \left (x^{2}\right ) - 3 \, x}{3 \, {\left (e \log \left (x^{2}\right ) - 3 \, e\right )}}\right )} \end {dmath*}
integrate((((4+4*x)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x )*log(x^2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*x*exp(1)+x^3)*log(x^2)-9* x*exp(1)-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*exp (1)*log(x^2)+9*exp(1)),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 4.71 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \begin {dmath*} \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 x e^{\frac {- 3 x^{3} - 9 e x + 3 x + \left (x^{3} + 3 e x\right ) \log {\left (x^{2} \right )}}{3 e \log {\left (x^{2} \right )} - 9 e}} \end {dmath*}
integrate((((4+4*x)*exp(1)+4*x**3)*ln(x**2)**2+((-24*x-24)*exp(1)-24*x**3+ 4*x)*ln(x**2)+(36*x+36)*exp(1)+36*x**3-20*x)*exp(((3*x*exp(1)+x**3)*ln(x** 2)-9*x*exp(1)-3*x**3+3*x)/(3*exp(1)*ln(x**2)-9*exp(1)))/(exp(1)*ln(x**2)** 2-6*exp(1)*ln(x**2)+9*exp(1)),x)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).
Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \begin {dmath*} \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 \, x e^{\left (\frac {2 \, x^{3} \log \left (x\right )}{3 \, {\left (2 \, e \log \left (x\right ) - 3 \, e\right )}} - \frac {x^{3}}{2 \, e \log \left (x\right ) - 3 \, e} + \frac {2 \, x \log \left (x\right )}{2 \, \log \left (x\right ) - 3} + \frac {x}{2 \, e \log \left (x\right ) - 3 \, e} - \frac {3 \, x}{2 \, \log \left (x\right ) - 3}\right )} \end {dmath*}
integrate((((4+4*x)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x )*log(x^2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*x*exp(1)+x^3)*log(x^2)-9* x*exp(1)-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*exp (1)*log(x^2)+9*exp(1)),x, algorithm=\
4*x*e^(2/3*x^3*log(x)/(2*e*log(x) - 3*e) - x^3/(2*e*log(x) - 3*e) + 2*x*lo g(x)/(2*log(x) - 3) + x/(2*e*log(x) - 3*e) - 3*x/(2*log(x) - 3))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 1.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59 \begin {dmath*} \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=4 \, x e^{\left (\frac {x^{3} \log \left (x^{2}\right ) - 3 \, x^{3} + 3 \, x e \log \left (x^{2}\right ) - 9 \, x e + 3 \, x}{3 \, {\left (e \log \left (x^{2}\right ) - 3 \, e\right )}}\right )} \end {dmath*}
integrate((((4+4*x)*exp(1)+4*x^3)*log(x^2)^2+((-24*x-24)*exp(1)-24*x^3+4*x )*log(x^2)+(36*x+36)*exp(1)+36*x^3-20*x)*exp(((3*x*exp(1)+x^3)*log(x^2)-9* x*exp(1)-3*x^3+3*x)/(3*exp(1)*log(x^2)-9*exp(1)))/(exp(1)*log(x^2)^2-6*exp (1)*log(x^2)+9*exp(1)),x, algorithm=\
Time = 15.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.97 \begin {dmath*} \int \frac {e^{\frac {3 x-9 e x-3 x^3+\left (3 e x+x^3\right ) \log \left (x^2\right )}{-9 e+3 e \log \left (x^2\right )}} \left (-20 x+36 x^3+e (36+36 x)+\left (e (-24-24 x)+4 x-24 x^3\right ) \log \left (x^2\right )+\left (4 x^3+e (4+4 x)\right ) \log ^2\left (x^2\right )\right )}{9 e-6 e \log \left (x^2\right )+e \log ^2\left (x^2\right )} \, dx=\frac {4\,x\,{\mathrm {e}}^{\frac {3\,x^3}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}\,{\mathrm {e}}^{\frac {9\,x\,\mathrm {e}}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}\,{\mathrm {e}}^{-\frac {3\,x}{9\,\mathrm {e}-3\,\ln \left (x^2\right )\,\mathrm {e}}}}{{\left (x^2\right )}^{\frac {x^3+3\,\mathrm {e}\,x}{3\,\left (3\,\mathrm {e}-\ln \left (x^2\right )\,\mathrm {e}\right )}}} \end {dmath*}
int((exp(-(3*x + log(x^2)*(3*x*exp(1) + x^3) - 9*x*exp(1) - 3*x^3)/(9*exp( 1) - 3*log(x^2)*exp(1)))*(log(x^2)^2*(4*x^3 + exp(1)*(4*x + 4)) - log(x^2) *(24*x^3 - 4*x + exp(1)*(24*x + 24)) - 20*x + 36*x^3 + exp(1)*(36*x + 36)) )/(9*exp(1) - 6*log(x^2)*exp(1) + log(x^2)^2*exp(1)),x)