Optimal. Leaf size=29 \[ \frac {\left (3+x \left (e^{e^x}-2 x+\log (x)\right )\right )^2}{\left (3 x+3 x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 7.09, 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 {-18-30 x+6 x^2+20 x^3+4 x^4+e^{2 e^x} \left (-2 x^3+e^x \left (2 x^3+2 x^4\right )\right )+\left (-6 x-16 x^2-2 x^3+4 x^4\right ) \log (x)-2 x^3 \log ^2(x)+e^{e^x} \left (-6 x-16 x^2-2 x^3+4 x^4+e^x \left (6 x^2+6 x^3-4 x^4-4 x^5\right )+\left (-4 x^3+e^x \left (2 x^3+2 x^4\right )\right ) \log (x)\right )}{9 x^3+27 x^4+27 x^5+9 x^6} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (3+e^{e^x} x-2 x^2+x \log (x)\right ) \left (-3-5 x+\left (-1-e^{e^x}+e^{e^x+x}\right ) x^2+e^{e^x+x} x^3-x^2 \log (x)\right )}{9 x^3 (1+x)^3} \, dx\\ &=\frac {2}{9} \int \frac {\left (3+e^{e^x} x-2 x^2+x \log (x)\right ) \left (-3-5 x+\left (-1-e^{e^x}+e^{e^x+x}\right ) x^2+e^{e^x+x} x^3-x^2 \log (x)\right )}{x^3 (1+x)^3} \, dx\\ &=\frac {2}{9} \int \left (\frac {3 \left (-3-e^{e^x} x+2 x^2-x \log (x)\right )}{x^3 (1+x)^3}+\frac {5 \left (-3-e^{e^x} x+2 x^2-x \log (x)\right )}{x^2 (1+x)^3}+\frac {-3-e^{e^x} x+2 x^2-x \log (x)}{x (1+x)^3}+\frac {\log (x) \left (-3-e^{e^x} x+2 x^2-x \log (x)\right )}{x (1+x)^3}-\frac {e^{e^x} \left (3+e^{e^x} x-2 x^2+x \log (x)\right )}{x (1+x)^3}+\frac {e^{e^x+x} \left (3+e^{e^x} x-2 x^2+x \log (x)\right )}{x (1+x)^2}\right ) \, dx\\ &=\frac {2}{9} \int \frac {-3-e^{e^x} x+2 x^2-x \log (x)}{x (1+x)^3} \, dx+\frac {2}{9} \int \frac {\log (x) \left (-3-e^{e^x} x+2 x^2-x \log (x)\right )}{x (1+x)^3} \, dx-\frac {2}{9} \int \frac {e^{e^x} \left (3+e^{e^x} x-2 x^2+x \log (x)\right )}{x (1+x)^3} \, dx+\frac {2}{9} \int \frac {e^{e^x+x} \left (3+e^{e^x} x-2 x^2+x \log (x)\right )}{x (1+x)^2} \, dx+\frac {2}{3} \int \frac {-3-e^{e^x} x+2 x^2-x \log (x)}{x^3 (1+x)^3} \, dx+\frac {10}{9} \int \frac {-3-e^{e^x} x+2 x^2-x \log (x)}{x^2 (1+x)^3} \, dx\\ &=\frac {2}{9} \int \left (-\frac {e^{e^x}}{(1+x)^3}-\frac {3}{x (1+x)^3}+\frac {2 x}{(1+x)^3}-\frac {\log (x)}{(1+x)^3}\right ) \, dx-\frac {2}{9} \int \left (\frac {e^{2 e^x}}{(1+x)^3}+\frac {3 e^{e^x}}{x (1+x)^3}-\frac {2 e^{e^x} x}{(1+x)^3}+\frac {e^{e^x} \log (x)}{(1+x)^3}\right ) \, dx+\frac {2}{9} \int \left (-\frac {e^{e^x} \log (x)}{(1+x)^3}-\frac {3 \log (x)}{x (1+x)^3}+\frac {2 x \log (x)}{(1+x)^3}-\frac {\log ^2(x)}{(1+x)^3}\right ) \, dx+\frac {2}{9} \int \left (\frac {e^{2 e^x+x}}{(1+x)^2}+\frac {e^{e^x+x} \left (3-2 x^2+x \log (x)\right )}{x (1+x)^2}\right ) \, dx+\frac {2}{3} \int \left (-\frac {3}{x^3 (1+x)^3}-\frac {e^{e^x}}{x^2 (1+x)^3}+\frac {2}{x (1+x)^3}-\frac {\log (x)}{x^2 (1+x)^3}\right ) \, dx+\frac {10}{9} \int \left (\frac {2}{(1+x)^3}-\frac {3}{x^2 (1+x)^3}-\frac {e^{e^x}}{x (1+x)^3}-\frac {\log (x)}{x (1+x)^3}\right ) \, dx\\ &=-\frac {10}{9 (1+x)^2}-\frac {2}{9} \int \frac {e^{e^x}}{(1+x)^3} \, dx-\frac {2}{9} \int \frac {e^{2 e^x}}{(1+x)^3} \, dx+\frac {2}{9} \int \frac {e^{2 e^x+x}}{(1+x)^2} \, dx-\frac {2}{9} \int \frac {\log (x)}{(1+x)^3} \, dx-2 \left (\frac {2}{9} \int \frac {e^{e^x} \log (x)}{(1+x)^3} \, dx\right )-\frac {2}{9} \int \frac {\log ^2(x)}{(1+x)^3} \, dx+\frac {2}{9} \int \frac {e^{e^x+x} \left (3-2 x^2+x \log (x)\right )}{x (1+x)^2} \, dx+\frac {4}{9} \int \frac {x}{(1+x)^3} \, dx+\frac {4}{9} \int \frac {e^{e^x} x}{(1+x)^3} \, dx+\frac {4}{9} \int \frac {x \log (x)}{(1+x)^3} \, dx-\frac {2}{3} \int \frac {e^{e^x}}{x^2 (1+x)^3} \, dx-\frac {2}{3} \int \frac {1}{x (1+x)^3} \, dx-\frac {2}{3} \int \frac {e^{e^x}}{x (1+x)^3} \, dx-\frac {2}{3} \int \frac {\log (x)}{x^2 (1+x)^3} \, dx-\frac {2}{3} \int \frac {\log (x)}{x (1+x)^3} \, dx-\frac {10}{9} \int \frac {e^{e^x}}{x (1+x)^3} \, dx-\frac {10}{9} \int \frac {\log (x)}{x (1+x)^3} \, dx+\frac {4}{3} \int \frac {1}{x (1+x)^3} \, dx-2 \int \frac {1}{x^3 (1+x)^3} \, dx-\frac {10}{3} \int \frac {1}{x^2 (1+x)^3} \, dx\\ &=-\frac {10}{9 (1+x)^2}+\frac {2 x^2}{9 (1+x)^2}+\frac {\log (x)}{9 (1+x)^2}+\frac {2 x^2 \log (x)}{9 (1+x)^2}+\frac {\log ^2(x)}{9 (1+x)^2}-\frac {1}{9} \int \frac {1}{x (1+x)^2} \, dx-\frac {2}{9} \int \frac {e^{e^x}}{(1+x)^3} \, dx-\frac {2}{9} \int \frac {e^{2 e^x}}{(1+x)^3} \, dx+\frac {2}{9} \int \frac {e^{2 e^x+x}}{(1+x)^2} \, dx-\frac {2}{9} \int \frac {x}{(1+x)^2} \, dx-\frac {2}{9} \int \frac {\log (x)}{x (1+x)^2} \, dx+\frac {2}{9} \int \left (\frac {e^{e^x+x} \left (3-2 x^2\right )}{x (1+x)^2}+\frac {e^{e^x+x} \log (x)}{(1+x)^2}\right ) \, dx+\frac {4}{9} \int \left (-\frac {e^{e^x}}{(1+x)^3}+\frac {e^{e^x}}{(1+x)^2}\right ) \, dx-\frac {2}{3} \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^3}-\frac {1}{(1+x)^2}\right ) \, dx-\frac {2}{3} \int \left (\frac {e^{e^x}}{-1-x}+\frac {e^{e^x}}{x}-\frac {e^{e^x}}{(1+x)^3}-\frac {e^{e^x}}{(1+x)^2}\right ) \, dx-\frac {2}{3} \int \left (\frac {e^{e^x}}{x^2}-\frac {3 e^{e^x}}{x}+\frac {e^{e^x}}{(1+x)^3}+\frac {2 e^{e^x}}{(1+x)^2}+\frac {3 e^{e^x}}{1+x}\right ) \, dx+\frac {2}{3} \int \frac {\log (x)}{(1+x)^3} \, dx-\frac {2}{3} \int \frac {\log (x)}{x (1+x)^2} \, dx-\frac {2}{3} \int \left (\frac {\log (x)}{x^2}-\frac {3 \log (x)}{x}+\frac {\log (x)}{(1+x)^3}+\frac {2 \log (x)}{(1+x)^2}+\frac {3 \log (x)}{1+x}\right ) \, dx-\frac {10}{9} \int \left (\frac {e^{e^x}}{-1-x}+\frac {e^{e^x}}{x}-\frac {e^{e^x}}{(1+x)^3}-\frac {e^{e^x}}{(1+x)^2}\right ) \, dx+\frac {10}{9} \int \frac {\log (x)}{(1+x)^3} \, dx-\frac {10}{9} \int \frac {\log (x)}{x (1+x)^2} \, dx+\frac {4}{3} \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^3}-\frac {1}{(1+x)^2}\right ) \, dx-2 \int \left (\frac {1}{x^3}-\frac {3}{x^2}+\frac {6}{x}-\frac {1}{(1+x)^3}-\frac {3}{(1+x)^2}-\frac {6}{1+x}\right ) \, dx-\frac {10}{3} \int \left (\frac {1}{x^2}-\frac {3}{x}+\frac {1}{(1+x)^3}+\frac {2}{(1+x)^2}+\frac {3}{1+x}\right ) \, dx-2 \left (-\left (\frac {2}{9} \int \frac {\int \frac {e^{e^x}}{(1+x)^3} \, dx}{x} \, dx\right )+\frac {1}{9} (2 \log (x)) \int \frac {e^{e^x}}{(1+x)^3} \, dx\right )\\ &=\frac {1}{x^2}-\frac {8}{3 x}-\frac {1}{9 (1+x)^2}+\frac {2 x^2}{9 (1+x)^2}+\frac {4}{3 (1+x)}-\frac {4 \log (x)}{3}-\frac {7 \log (x)}{9 (1+x)^2}+\frac {2 x^2 \log (x)}{9 (1+x)^2}+\frac {\log ^2(x)}{9 (1+x)^2}+\frac {4}{3} \log (1+x)-\frac {1}{9} \int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx-\frac {2}{9} \int \frac {e^{e^x}}{(1+x)^3} \, dx-\frac {2}{9} \int \frac {e^{2 e^x}}{(1+x)^3} \, dx+\frac {2}{9} \int \frac {e^{2 e^x+x}}{(1+x)^2} \, dx+\frac {2}{9} \int \frac {e^{e^x+x} \left (3-2 x^2\right )}{x (1+x)^2} \, dx-\frac {2}{9} \int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx+\frac {2}{9} \int \frac {\log (x)}{(1+x)^2} \, dx+\frac {2}{9} \int \frac {e^{e^x+x} \log (x)}{(1+x)^2} \, dx-\frac {2}{9} \int \frac {\log (x)}{x (1+x)} \, dx+\frac {1}{3} \int \frac {1}{x (1+x)^2} \, dx-\frac {4}{9} \int \frac {e^{e^x}}{(1+x)^3} \, dx+\frac {4}{9} \int \frac {e^{e^x}}{(1+x)^2} \, dx+\frac {5}{9} \int \frac {1}{x (1+x)^2} \, dx-\frac {2}{3} \int \frac {e^{e^x}}{-1-x} \, dx-\frac {2}{3} \int \frac {e^{e^x}}{x^2} \, dx-\frac {2}{3} \int \frac {e^{e^x}}{x} \, dx+\frac {2}{3} \int \frac {e^{e^x}}{(1+x)^2} \, dx-\frac {2}{3} \int \frac {\log (x)}{x^2} \, dx-\frac {2}{3} \int \frac {\log (x)}{(1+x)^3} \, dx+\frac {2}{3} \int \frac {\log (x)}{(1+x)^2} \, dx-\frac {2}{3} \int \frac {\log (x)}{x (1+x)} \, dx-\frac {10}{9} \int \frac {e^{e^x}}{-1-x} \, dx-\frac {10}{9} \int \frac {e^{e^x}}{x} \, dx+\frac {10}{9} \int \frac {e^{e^x}}{(1+x)^3} \, dx+\frac {10}{9} \int \frac {e^{e^x}}{(1+x)^2} \, dx+\frac {10}{9} \int \frac {\log (x)}{(1+x)^2} \, dx-\frac {10}{9} \int \frac {\log (x)}{x (1+x)} \, dx-\frac {4}{3} \int \frac {e^{e^x}}{(1+x)^2} \, dx-\frac {4}{3} \int \frac {\log (x)}{(1+x)^2} \, dx+2 \int \frac {e^{e^x}}{x} \, dx-2 \int \frac {e^{e^x}}{1+x} \, dx+2 \int \frac {\log (x)}{x} \, dx-2 \int \frac {\log (x)}{1+x} \, dx-2 \left (-\left (\frac {2}{9} \int \frac {\int \frac {e^{e^x}}{(1+x)^3} \, dx}{x} \, dx\right )+\frac {1}{9} (2 \log (x)) \int \frac {e^{e^x}}{(1+x)^3} \, dx\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(78\) vs. \(2(29)=58\).
time = 0.32, size = 78, normalized size = 2.69 \begin {gather*} \frac {9-16 x^2+e^{2 e^x} x^2-8 x^3+e^{e^x} \left (6 x-4 x^3\right )-2 x \left (-3-e^{e^x} x+2 x^2\right ) \log (x)+x^2 \log ^2(x)}{9 x^2 (1+x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs.
\(2(27)=54\).
time = 0.17, size = 116, normalized size = 4.00
method | result | size |
risch | \(\frac {\ln \left (x \right )^{2}}{9 x^{2}+18 x +9}-\frac {2 \left (2 x^{2}-3\right ) \ln \left (x \right )}{9 x \left (x^{2}+2 x +1\right )}-\frac {8 x^{3}+16 x^{2}-9}{9 \left (x^{2}+2 x +1\right ) x^{2}}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}}}{9 x^{2}+18 x +9}-\frac {2 \left (2 x^{2}-x \ln \left (x \right )-3\right ) {\mathrm e}^{{\mathrm e}^{x}}}{9 x \left (x^{2}+2 x +1\right )}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs.
\(2 (28) = 56\).
time = 0.35, size = 167, normalized size = 5.76 \begin {gather*} -\frac {12 \, x^{3} + 18 \, x^{2} + 4 \, x - 1}{x^{4} + 2 \, x^{3} + x^{2}} + \frac {x \log \left (x\right )^{2} + 8 \, x^{2} + x e^{\left (2 \, e^{x}\right )} - 2 \, {\left (2 \, x^{2} - x \log \left (x\right ) - 3\right )} e^{\left (e^{x}\right )} - 2 \, {\left (2 \, x^{2} - 3\right )} \log \left (x\right ) + 14 \, x + 6}{9 \, {\left (x^{3} + 2 \, x^{2} + x\right )}} + \frac {5 \, {\left (6 \, x^{2} + 9 \, x + 2\right )}}{3 \, {\left (x^{3} + 2 \, x^{2} + x\right )}} + \frac {2 \, x + 3}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {2 \, {\left (2 \, x + 1\right )}}{9 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {10}{9 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs.
\(2 (28) = 56\).
time = 0.35, size = 79, normalized size = 2.72 \begin {gather*} \frac {x^{2} \log \left (x\right )^{2} - 8 \, x^{3} + x^{2} e^{\left (2 \, e^{x}\right )} - 16 \, x^{2} - 2 \, {\left (2 \, x^{3} - x^{2} \log \left (x\right ) - 3 \, x\right )} e^{\left (e^{x}\right )} - 2 \, {\left (2 \, x^{3} - 3 \, x\right )} \log \left (x\right ) + 9}{9 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs.
\(2 (26) = 52\).
time = 0.31, size = 158, normalized size = 5.45 \begin {gather*} \frac {\left (6 - 4 x^{2}\right ) \log {\left (x \right )}}{9 x^{3} + 18 x^{2} + 9 x} + \frac {\left (9 x^{3} + 18 x^{2} + 9 x\right ) e^{2 e^{x}} + \left (- 36 x^{4} + 18 x^{3} \log {\left (x \right )} - 72 x^{3} + 36 x^{2} \log {\left (x \right )} + 18 x^{2} + 18 x \log {\left (x \right )} + 108 x + 54\right ) e^{e^{x}}}{81 x^{5} + 324 x^{4} + 486 x^{3} + 324 x^{2} + 81 x} + \frac {- 8 x^{3} - 16 x^{2} + 9}{9 x^{4} + 18 x^{3} + 9 x^{2}} + \frac {\log {\left (x \right )}^{2}}{9 x^{2} + 18 x + 9} \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*} \text {could not integrate} \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 {30\,x+\ln \left (x\right )\,\left (-4\,x^4+2\,x^3+16\,x^2+6\,x\right )+2\,x^3\,{\ln \left (x\right )}^2-{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left ({\mathrm {e}}^x\,\left (2\,x^4+2\,x^3\right )-2\,x^3\right )-6\,x^2-20\,x^3-4\,x^4+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (6\,x-{\mathrm {e}}^x\,\left (-4\,x^5-4\,x^4+6\,x^3+6\,x^2\right )+16\,x^2+2\,x^3-4\,x^4-\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (2\,x^4+2\,x^3\right )-4\,x^3\right )\right )+18}{9\,x^6+27\,x^5+27\,x^4+9\,x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________