Integrand size = 73, antiderivative size = 22 \[ \int \frac {-3+e^x \left (3 x+6 x^2+3 x^3\right )}{\left (2 e^{e^x \left (1+x^2\right )}-x\right ) \log \left (e^{-e^x \left (1+x^2\right )} \left (-12 e^{e^x \left (1+x^2\right )}+6 x\right )\right )} \, dx=3 \log \left (\log \left (6 \left (-2+e^{-e^x \left (1+x^2\right )} x\right )\right )\right ) \]
Time = 0.44 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-3+e^x \left (3 x+6 x^2+3 x^3\right )}{\left (2 e^{e^x \left (1+x^2\right )}-x\right ) \log \left (e^{-e^x \left (1+x^2\right )} \left (-12 e^{e^x \left (1+x^2\right )}+6 x\right )\right )} \, dx=3 \log \left (\log \left (-12+6 e^{-e^x \left (1+x^2\right )} x\right )\right ) \]
Integrate[(-3 + E^x*(3*x + 6*x^2 + 3*x^3))/((2*E^(E^x*(1 + x^2)) - x)*Log[ (-12*E^(E^x*(1 + x^2)) + 6*x)/E^(E^x*(1 + x^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.
\(\displaystyle \int \frac {e^x \left (3 x^3+6 x^2+3 x\right )-3}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (e^{-e^x \left (x^2+1\right )} \left (6 x-12 e^{e^x \left (x^2+1\right )}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^x \left (3 x^3+6 x^2+3 x\right )-3}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (-6 e^{-e^x \left (x^2+1\right )} \left (2 e^{e^x \left (x^2+1\right )}-x\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {6 e^x x^2}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (6 e^{-e^x \left (x^2+1\right )} x-12\right )}+\frac {3 e^x x}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (6 e^{-e^x \left (x^2+1\right )} x-12\right )}-\frac {3}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (6 e^{-e^x \left (x^2+1\right )} x-12\right )}+\frac {3 e^x x^3}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (6 e^{-e^x \left (x^2+1\right )} x-12\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \int \frac {1}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (6 e^{-e^x \left (x^2+1\right )} x-12\right )}dx+3 \int \frac {e^x x}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (6 e^{-e^x \left (x^2+1\right )} x-12\right )}dx+6 \int \frac {e^x x^2}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (6 e^{-e^x \left (x^2+1\right )} x-12\right )}dx+3 \int \frac {e^x x^3}{\left (2 e^{e^x \left (x^2+1\right )}-x\right ) \log \left (6 e^{-e^x \left (x^2+1\right )} x-12\right )}dx\) |
Int[(-3 + E^x*(3*x + 6*x^2 + 3*x^3))/((2*E^(E^x*(1 + x^2)) - x)*Log[(-12*E ^(E^x*(1 + x^2)) + 6*x)/E^(E^x*(1 + x^2))]),x]
3.24.38.3.1 Defintions of rubi rules used
Time = 0.38 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50
method | result | size |
parallelrisch | \(3 \ln \left (\ln \left (-6 \left (2 \,{\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}-x \right ) {\mathrm e}^{-\left (x^{2}+1\right ) {\mathrm e}^{x}}\right )\right )\) | \(33\) |
risch | \(3 \ln \left (\ln \left ({\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (-2 \,{\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\left (x^{2}+1\right ) {\mathrm e}^{x}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\left (x^{2}+1\right ) {\mathrm e}^{x}} \left (-2 \,{\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}+x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (-2 \,{\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\left (x^{2}+1\right ) {\mathrm e}^{x}} \left (-2 \,{\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}+x \right )\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-\left (x^{2}+1\right ) {\mathrm e}^{x}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\left (x^{2}+1\right ) {\mathrm e}^{x}} \left (-2 \,{\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}+x \right )\right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-\left (x^{2}+1\right ) {\mathrm e}^{x}} \left (-2 \,{\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}+x \right )\right )^{3}}{2}-\ln \left (2\right )-\ln \left (3\right )-\ln \left (-2 \,{\mathrm e}^{\left (x^{2}+1\right ) {\mathrm e}^{x}}+x \right )\right )\) | \(231\) |
int(((3*x^3+6*x^2+3*x)*exp(x)-3)/(2*exp((x^2+1)*exp(x))-x)/ln((-12*exp((x^ 2+1)*exp(x))+6*x)/exp((x^2+1)*exp(x))),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-3+e^x \left (3 x+6 x^2+3 x^3\right )}{\left (2 e^{e^x \left (1+x^2\right )}-x\right ) \log \left (e^{-e^x \left (1+x^2\right )} \left (-12 e^{e^x \left (1+x^2\right )}+6 x\right )\right )} \, dx=3 \, \log \left (\log \left (6 \, {\left (x - 2 \, e^{\left ({\left (x^{2} + 1\right )} e^{x}\right )}\right )} e^{\left (-{\left (x^{2} + 1\right )} e^{x}\right )}\right )\right ) \]
integrate(((3*x^3+6*x^2+3*x)*exp(x)-3)/(2*exp((x^2+1)*exp(x))-x)/log((-12* exp((x^2+1)*exp(x))+6*x)/exp((x^2+1)*exp(x))),x, algorithm=\
Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-3+e^x \left (3 x+6 x^2+3 x^3\right )}{\left (2 e^{e^x \left (1+x^2\right )}-x\right ) \log \left (e^{-e^x \left (1+x^2\right )} \left (-12 e^{e^x \left (1+x^2\right )}+6 x\right )\right )} \, dx=3 \log {\left (\log {\left (\left (6 x - 12 e^{\left (x^{2} + 1\right ) e^{x}}\right ) e^{- \left (x^{2} + 1\right ) e^{x}} \right )} \right )} \]
integrate(((3*x**3+6*x**2+3*x)*exp(x)-3)/(2*exp((x**2+1)*exp(x))-x)/ln((-1 2*exp((x**2+1)*exp(x))+6*x)/exp((x**2+1)*exp(x))),x)
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {-3+e^x \left (3 x+6 x^2+3 x^3\right )}{\left (2 e^{e^x \left (1+x^2\right )}-x\right ) \log \left (e^{-e^x \left (1+x^2\right )} \left (-12 e^{e^x \left (1+x^2\right )}+6 x\right )\right )} \, dx=3 \, \log \left (i \, \pi - {\left (x^{2} + 1\right )} e^{x} + \log \left (3\right ) + \log \left (2\right ) + \log \left (-x + 2 \, e^{\left (x^{2} e^{x} + e^{x}\right )}\right )\right ) \]
integrate(((3*x^3+6*x^2+3*x)*exp(x)-3)/(2*exp((x^2+1)*exp(x))-x)/log((-12* exp((x^2+1)*exp(x))+6*x)/exp((x^2+1)*exp(x))),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-3+e^x \left (3 x+6 x^2+3 x^3\right )}{\left (2 e^{e^x \left (1+x^2\right )}-x\right ) \log \left (e^{-e^x \left (1+x^2\right )} \left (-12 e^{e^x \left (1+x^2\right )}+6 x\right )\right )} \, dx=3 \, \log \left (\log \left (6 \, {\left (x - 2 \, e^{\left (x^{2} e^{x} + e^{x}\right )}\right )} e^{\left (-x^{2} e^{x} - e^{x}\right )}\right )\right ) \]
integrate(((3*x^3+6*x^2+3*x)*exp(x)-3)/(2*exp((x^2+1)*exp(x))-x)/log((-12* exp((x^2+1)*exp(x))+6*x)/exp((x^2+1)*exp(x))),x, algorithm=\
Time = 9.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-3+e^x \left (3 x+6 x^2+3 x^3\right )}{\left (2 e^{e^x \left (1+x^2\right )}-x\right ) \log \left (e^{-e^x \left (1+x^2\right )} \left (-12 e^{e^x \left (1+x^2\right )}+6 x\right )\right )} \, dx=3\,\ln \left (\ln \left (6\,x\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-{\mathrm {e}}^x}-12\right )\right ) \]