Integrand size = 144, antiderivative size = 36 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=\frac {-3+\frac {x}{2-\frac {\log (3)}{4 e^3 \left (-e^{2 e^3}+e^x\right )}}}{x} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(164\) vs. \(2(36)=72\).
Time = 1.18 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.56, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6, 6873, 6874, 2320, 46, 36, 29, 31} \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=-\frac {3}{x}+\frac {x \log (3)}{16 e^{3+2 e^3}+\log (9)}-\frac {x \log (3)}{2 \left (8 e^{3+2 e^3}+\log (3)\right )}+\frac {\log (3) \log \left (-8 e^{x+3}+8 e^{3+2 e^3}+\log (3)\right )}{2 \left (8 e^{3+2 e^3}+\log (3)\right )}-\frac {\log (3) \log \left (-16 e^{x+3}+16 e^{3+2 e^3}+\log (9)\right )}{16 e^{3+2 e^3}+\log (9)}-\frac {\log (3)}{2 \left (-8 e^{x+3}+8 e^{3+2 e^3}+\log (3)\right )} \]
[In]
[Out]
Rule 6
Rule 29
Rule 31
Rule 36
Rule 46
Rule 2320
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )+x^2 \left (64 e^{6+4 e^3}+\log ^2(3)\right )} \, dx \\ & = \int \frac {192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )+192 e^{6+4 e^3} \left (1+\frac {1}{64} e^{-6-4 e^3} \log ^2(3)\right )}{x^2 \left (8 e^{3+x}-8 e^{3+2 e^3} \left (1+\frac {1}{8} e^{-3-2 e^3} \log (3)\right )\right )^2} \, dx \\ & = \int \left (\frac {3}{x^2}+\frac {\left (-8 e^{3+2 e^3}-\log (3)\right ) \log (3)}{2 \left (8 e^{3+x}-8 e^{3+2 e^3} \left (1+\frac {1}{8} e^{-3-2 e^3} \log (3)\right )\right )^2}+\frac {\log (3)}{-16 e^{3+x}+16 e^{3+2 e^3} \left (1+\frac {1}{8} e^{-3-2 e^3} \log (3)\right )}\right ) \, dx \\ & = -\frac {3}{x}+\log (3) \int \frac {1}{-16 e^{3+x}+16 e^{3+2 e^3} \left (1+\frac {1}{8} e^{-3-2 e^3} \log (3)\right )} \, dx-\frac {1}{2} \left (\log (3) \left (8 e^{3+2 e^3}+\log (3)\right )\right ) \int \frac {1}{\left (8 e^{3+x}-8 e^{3+2 e^3} \left (1+\frac {1}{8} e^{-3-2 e^3} \log (3)\right )\right )^2} \, dx \\ & = -\frac {3}{x}+\log (3) \text {Subst}\left (\int \frac {1}{x \left (16 e^{3+2 e^3}-16 x+\log (9)\right )} \, dx,x,e^{3+x}\right )-\frac {1}{2} \left (\log (3) \left (8 e^{3+2 e^3}+\log (3)\right )\right ) \text {Subst}\left (\int \frac {1}{x \left (8 e^{3+2 e^3}-8 x+\log (3)\right )^2} \, dx,x,e^{3+x}\right ) \\ & = -\frac {3}{x}-\frac {1}{2} \left (\log (3) \left (8 e^{3+2 e^3}+\log (3)\right )\right ) \text {Subst}\left (\int \left (\frac {1}{x \left (8 e^{3+2 e^3}+\log (3)\right )^2}+\frac {8}{\left (8 e^{3+2 e^3}+\log (3)\right ) \left (8 e^{3+2 e^3}-8 x+\log (3)\right )^2}+\frac {8}{\left (8 e^{3+2 e^3}+\log (3)\right )^2 \left (8 e^{3+2 e^3}-8 x+\log (3)\right )}\right ) \, dx,x,e^{3+x}\right )+\frac {\log (3) \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{3+x}\right )}{16 e^{3+2 e^3}+\log (9)}+\frac {(16 \log (3)) \text {Subst}\left (\int \frac {1}{16 e^{3+2 e^3}-16 x+\log (9)} \, dx,x,e^{3+x}\right )}{16 e^{3+2 e^3}+\log (9)} \\ & = -\frac {3}{x}-\frac {x \log (3)}{2 \left (8 e^{3+2 e^3}+\log (3)\right )}-\frac {\log (3)}{2 \left (8 e^{3+2 e^3}-8 e^{3+x}+\log (3)\right )}+\frac {x \log (3)}{16 e^{3+2 e^3}+\log (9)}+\frac {\log (3) \log \left (8 e^{3+2 e^3}-8 e^{3+x}+\log (3)\right )}{2 \left (8 e^{3+2 e^3}+\log (3)\right )}-\frac {\log (3) \log \left (16 e^{3+2 e^3}-16 e^{3+x}+\log (9)\right )}{16 e^{3+2 e^3}+\log (9)} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=-\frac {3}{x}-\frac {\log (3)}{16 e^{3+2 e^3}-16 e^{3+x}+\log (9)} \]
[In]
[Out]
Time = 5.68 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-\frac {3}{x}-\frac {\ln \left (3\right )}{2 \left (8 \,{\mathrm e}^{3+2 \,{\mathrm e}^{3}}-8 \,{\mathrm e}^{3+x}+\ln \left (3\right )\right )}\) | \(31\) |
norman | \(\frac {-\frac {x \ln \left (3\right )}{2}+24 \,{\mathrm e}^{x} {\mathrm e}^{3}-24 \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{3}}-3 \ln \left (3\right )}{x \left (8 \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{3}}-8 \,{\mathrm e}^{x} {\mathrm e}^{3}+\ln \left (3\right )\right )}\) | \(50\) |
parallelrisch | \(-\frac {\left (192 \,{\mathrm e}^{6} {\mathrm e}^{2 \,{\mathrm e}^{3}}+4 x \,{\mathrm e}^{3} \ln \left (3\right )-192 \,{\mathrm e}^{6} {\mathrm e}^{x}+24 \,{\mathrm e}^{3} \ln \left (3\right )\right ) {\mathrm e}^{-3}}{8 \left (8 \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{3}}-8 \,{\mathrm e}^{x} {\mathrm e}^{3}+\ln \left (3\right )\right ) x}\) | \(63\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=-\frac {{\left (x + 6\right )} e^{3} \log \left (3\right ) - 48 \, e^{\left (x + 6\right )} + 48 \, e^{\left (2 \, e^{3} + 6\right )}}{2 \, {\left (x e^{3} \log \left (3\right ) - 8 \, x e^{\left (x + 6\right )} + 8 \, x e^{\left (2 \, e^{3} + 6\right )}\right )}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=\frac {\log {\left (3 \right )}}{16 e^{3} e^{x} - 16 e^{3} e^{2 e^{3}} - 2 \log {\left (3 \right )}} - \frac {3}{x} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=-\frac {x \log \left (3\right ) - 48 \, e^{\left (x + 3\right )} + 48 \, e^{\left (2 \, e^{3} + 3\right )} + 6 \, \log \left (3\right )}{2 \, {\left (x {\left (8 \, e^{\left (2 \, e^{3} + 3\right )} + \log \left (3\right )\right )} - 8 \, x e^{\left (x + 3\right )}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (31) = 62\).
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.17 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=\frac {{\left (x + 3\right )} \log \left (3\right ) - 48 \, e^{\left (x + 3\right )} + 48 \, e^{\left (2 \, e^{3} + 3\right )} + 3 \, \log \left (3\right )}{2 \, {\left (8 \, {\left (x + 3\right )} e^{\left (x + 3\right )} - 8 \, {\left (x + 3\right )} e^{\left (2 \, e^{3} + 3\right )} - {\left (x + 3\right )} \log \left (3\right ) - 24 \, e^{\left (x + 3\right )} + 24 \, e^{\left (2 \, e^{3} + 3\right )} + 3 \, \log \left (3\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=\int \frac {192\,{\mathrm {e}}^{4\,{\mathrm {e}}^3+6}+192\,{\mathrm {e}}^{2\,x+6}-{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,\left (384\,{\mathrm {e}}^{x+6}-48\,{\mathrm {e}}^3\,\ln \left (3\right )\right )+3\,{\ln \left (3\right )}^2-{\mathrm {e}}^{x+3}\,\ln \left (3\right )\,\left (4\,x^2+48\right )}{x^2\,{\ln \left (3\right )}^2-{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,\left (128\,x^2\,{\mathrm {e}}^{x+6}-16\,x^2\,{\mathrm {e}}^3\,\ln \left (3\right )\right )+64\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^3+6}+64\,x^2\,{\mathrm {e}}^{2\,x+6}-16\,x^2\,{\mathrm {e}}^{x+3}\,\ln \left (3\right )} \,d x \]
[In]
[Out]