Optimal. Leaf size=35 \[ 1+\log \left (1-x-e^4 x+\frac {3}{\frac {x}{1-e^x x^2}+\log (3)}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 12.42, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+\left (1+e^4\right ) x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx\\ &=\int \frac {3+\left (1+e^4\right ) x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+\left (1+e^4\right ) x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx\\ &=\int \frac {-\left (\left (1+e^4\right ) x^2\right )-\left (2 x+2 e^4 x\right ) \log (3)-e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)-3 \left (1+\frac {1}{3} \left (1+e^4\right ) \log ^2(3)\right )-e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{3 x+x^2-\left (1+e^4\right ) x^3-\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)-\left (-1+x+e^4 x\right ) \log ^2(3)-e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )-e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx\\ &=\int \left (\frac {\left (-1-e^4\right ) \log (3)}{3+\log (3)-\left (1+e^4\right ) x \log (3)}+\frac {x^2+x (1+\log (3))+\log (9)}{x \left (x+\log (3)-e^x x^2 \log (3)\right )}+\frac {\left (1+e^4\right )^2 x^4 \log (3)+2 (3+\log (3))^2+x^2 \left (3-7 \log (3)-2 e^4 (4-\log (3)) \log (3)+2 e^8 \log ^2(3)\right )-\left (1+e^4\right ) x^3 \left (3+\log (3)-\log ^2(3)-e^4 \log (3) (1+\log (3))\right )+x \left (12-3 \log ^2(3)-4 e^4 \log (3) (3+\log (3))-\log (243)\right )}{x \left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right ) \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )}\right ) \, dx\\ &=\log \left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right )+\int \frac {x^2+x (1+\log (3))+\log (9)}{x \left (x+\log (3)-e^x x^2 \log (3)\right )} \, dx+\int \frac {\left (1+e^4\right )^2 x^4 \log (3)+2 (3+\log (3))^2+x^2 \left (3-7 \log (3)-2 e^4 (4-\log (3)) \log (3)+2 e^8 \log ^2(3)\right )-\left (1+e^4\right ) x^3 \left (3+\log (3)-\log ^2(3)-e^4 \log (3) (1+\log (3))\right )+x \left (12-3 \log ^2(3)-4 e^4 \log (3) (3+\log (3))-\log (243)\right )}{x \left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right ) \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )} \, dx\\ &=\log \left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right )+\int \left (-\frac {x}{-x-\log (3)+e^x x^2 \log (3)}-\frac {1+\log (3)}{-x-\log (3)+e^x x^2 \log (3)}-\frac {\log (9)}{x \left (-x-\log (3)+e^x x^2 \log (3)\right )}\right ) \, dx+\int \left (\frac {\left (-1-e^4\right ) x^2}{\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )}+\frac {2 (3+\log (3))}{x \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )}+\frac {-3+4 \log (3)-\log ^2(3)-2 e^4 \log ^2(3)}{\log (3) \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )}+\frac {x \left (-\log (3)-e^4 (1+\log (3))\right )}{\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )}+\frac {2 \left (1+e^4\right ) \log (3) (3+\log (3)) \left (1-\frac {-9+\log ^2(3)+2 \log ^3(3)+2 e^4 \log ^2(3) (3+\log (3))-\log (27)+\log (3) \log (243)}{2 \left (1+e^4\right ) \log ^2(3) (3+\log (3))}\right )}{\left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right ) \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )}\right ) \, dx\\ &=\log \left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right )+\left (-1-e^4\right ) \int \frac {x^2}{\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )} \, dx+(-1-\log (3)) \int \frac {1}{-x-\log (3)+e^x x^2 \log (3)} \, dx+(2 (3+\log (3))) \int \frac {1}{x \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )} \, dx+\frac {\left (-3+4 \log (3)-\log ^2(3)-2 e^4 \log ^2(3)\right ) \int \frac {1}{\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )} \, dx}{\log (3)}+\left (-\log (3)-e^4 (1+\log (3))\right ) \int \frac {x}{\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )} \, dx-\log (9) \int \frac {1}{x \left (-x-\log (3)+e^x x^2 \log (3)\right )} \, dx+\frac {\left (9+5 \log ^2(3)+\log (27)-\log (3) \log (243)\right ) \int \frac {1}{\left (3+\log (3)-\left (1+e^4\right ) x \log (3)\right ) \left (\left (1+e^4\right ) x^2-3 \left (1+\frac {\log (3)}{3}\right )+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-e^x \left (1+e^4\right ) x^3 \log (3)-x \left (1-\left (1+e^4\right ) \log (3)\right )\right )} \, dx}{\log (3)}-\int \frac {x}{-x-\log (3)+e^x x^2 \log (3)} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [F] time = 13.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.85, size = 118, normalized size = 3.37 \begin {gather*} \log \left ({\left (x e^{4} + x - 1\right )} \log \relax (3) - 3\right ) + \log \left (\frac {x^{2} e^{4} + x^{2} + {\left (3 \, x^{2} - {\left (x^{3} e^{4} + x^{3} - x^{2}\right )} \log \relax (3)\right )} e^{x} + {\left (x e^{4} + x - 1\right )} \log \relax (3) - x - 3}{3 \, x^{2} - {\left (x^{3} e^{4} + x^{3} - x^{2}\right )} \log \relax (3)}\right ) - \log \left (\frac {x^{2} e^{x} \log \relax (3) - x - \log \relax (3)}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.89, size = 85, normalized size = 2.43
method | result | size |
norman | \(-\ln \left (x^{2} \ln \relax (3) {\mathrm e}^{x}-\ln \relax (3)-x \right )+\ln \left ({\mathrm e}^{4} {\mathrm e}^{x} \ln \relax (3) x^{3}+{\mathrm e}^{x} \ln \relax (3) x^{3}-x^{2} \ln \relax (3) {\mathrm e}^{x}-{\mathrm e}^{4} x \ln \relax (3)-x^{2} {\mathrm e}^{4}-3 \,{\mathrm e}^{x} x^{2}-x \ln \relax (3)-x^{2}+\ln \relax (3)+x +3\right )\) | \(85\) |
risch | \(\ln \left (\left ({\mathrm e}^{4} \ln \relax (3)+\ln \relax (3)\right ) x -\ln \relax (3)-3\right )+\ln \left ({\mathrm e}^{x}-\frac {{\mathrm e}^{4} x \ln \relax (3)+x^{2} {\mathrm e}^{4}+x \ln \relax (3)+x^{2}-\ln \relax (3)-x -3}{x^{2} \left ({\mathrm e}^{4} x \ln \relax (3)+x \ln \relax (3)-\ln \relax (3)-3\right )}\right )-\ln \left ({\mathrm e}^{x}-\frac {\ln \relax (3)+x}{\ln \relax (3) x^{2}}\right )\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.79, size = 123, normalized size = 3.51 \begin {gather*} \log \left ({\left (e^{4} \log \relax (3) + \log \relax (3)\right )} x - \log \relax (3) - 3\right ) + \log \left (-\frac {x^{2} {\left (e^{4} + 1\right )} + {\left (e^{4} \log \relax (3) + \log \relax (3) - 1\right )} x - {\left ({\left (e^{4} \log \relax (3) + \log \relax (3)\right )} x^{3} - x^{2} {\left (\log \relax (3) + 3\right )}\right )} e^{x} - \log \relax (3) - 3}{{\left (e^{4} \log \relax (3) + \log \relax (3)\right )} x^{3} - x^{2} {\left (\log \relax (3) + 3\right )}}\right ) - \log \left (\frac {x^{2} e^{x} \log \relax (3) - x - \log \relax (3)}{x^{2} \log \relax (3)}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {\ln \relax (3)\,\left (2\,x+2\,x\,{\mathrm {e}}^4\right )-{\mathrm {e}}^x\,\left ({\ln \relax (3)}^2\,\left (2\,x^2\,{\mathrm {e}}^4+2\,x^2\right )-3\,x^2-3\,x^3+\ln \relax (3)\,\left (2\,x^3\,{\mathrm {e}}^4+2\,x^3\right )\right )+x^2\,{\mathrm {e}}^4+x^2+{\ln \relax (3)}^2\,\left ({\mathrm {e}}^4+1\right )+{\mathrm {e}}^{2\,x}\,{\ln \relax (3)}^2\,\left (x^4\,{\mathrm {e}}^4+x^4\right )+3}{3\,x-{\ln \relax (3)}^2\,\left (x+x\,{\mathrm {e}}^4-1\right )-x^3\,{\mathrm {e}}^4+{\mathrm {e}}^x\,\left (\ln \relax (3)\,\left (2\,x^4\,{\mathrm {e}}^4-6\,x^2-2\,x^3+2\,x^4\right )+{\ln \relax (3)}^2\,\left (2\,x^3\,{\mathrm {e}}^4-2\,x^2+2\,x^3\right )-3\,x^3\right )-{\mathrm {e}}^{2\,x}\,\left ({\ln \relax (3)}^2\,\left (x^5\,{\mathrm {e}}^4-x^4+x^5\right )-3\,x^4\,\ln \relax (3)\right )+\ln \relax (3)\,\left (2\,x-2\,x^2\,{\mathrm {e}}^4-2\,x^2+3\right )+x^2-x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________