Integrand size = 69, antiderivative size = 23 \[ \int \frac {e^{\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )-x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (e^x (-6+x) \left (i \pi +\log \left (\frac {13}{6}\right )\right )+5 x \left (i \pi +\log \left (\frac {13}{6}\right )\right )\right )}{x^7} \, dx=e^{\frac {\left (e^x-x\right ) \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \] Output:
exp((ln(13/6)+I*Pi)/x^6*(exp(x)-x))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(23)=46\).
Time = 2.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {e^{\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )-x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (e^x (-6+x) \left (i \pi +\log \left (\frac {13}{6}\right )\right )+5 x \left (i \pi +\log \left (\frac {13}{6}\right )\right )\right )}{x^7} \, dx=-\frac {i \left (\frac {6}{13}\right )^{\frac {1}{x^5}} e^{-\frac {i \pi }{x^5}+\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{\pi -i \log \left (\frac {13}{6}\right )} \] Input:
Integrate[(E^((E^x*(I*Pi + Log[13/6]) - x*(I*Pi + Log[13/6]))/x^6)*(E^x*(- 6 + x)*(I*Pi + Log[13/6]) + 5*x*(I*Pi + Log[13/6])))/x^7,x]
Output:
((-I)*(6/13)^x^(-5)*E^(((-I)*Pi)/x^5 + (E^x*(I*Pi + Log[13/6]))/x^6)*(I*Pi + Log[13/6]))/(Pi - I*Log[13/6])
Time = 1.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7257}
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.
\(\displaystyle \int \frac {\left (e^x (x-6) \left (\log \left (\frac {13}{6}\right )+i \pi \right )+5 x \left (\log \left (\frac {13}{6}\right )+i \pi \right )\right ) \exp \left (\frac {e^x \left (\log \left (\frac {13}{6}\right )+i \pi \right )-x \left (\log \left (\frac {13}{6}\right )+i \pi \right )}{x^6}\right )}{x^7} \, dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle \exp \left (\frac {e^x \left (\log \left (\frac {13}{6}\right )+i \pi \right )-x \left (\log \left (\frac {13}{6}\right )+i \pi \right )}{x^6}\right )\) |
Input:
Int[(E^((E^x*(I*Pi + Log[13/6]) - x*(I*Pi + Log[13/6]))/x^6)*(E^x*(-6 + x) *(I*Pi + Log[13/6]) + 5*x*(I*Pi + Log[13/6])))/x^7,x]
Output:
E^((E^x*(I*Pi + Log[13/6]) - x*(I*Pi + Log[13/6]))/x^6)
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 21.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (\ln \left (\frac {13}{6}\right )+i \pi \right ) \left ({\mathrm e}^{x}-x \right )}{x^{6}}}\) | \(19\) |
norman | \({\mathrm e}^{\frac {\left (\ln \left (\frac {13}{6}\right )+i \pi \right ) {\mathrm e}^{x}-x \left (\ln \left (\frac {13}{6}\right )+i \pi \right )}{x^{6}}}\) | \(27\) |
risch | \({\mathrm e}^{\frac {\left (\ln \left (13\right )-\ln \left (2\right )-\ln \left (3\right )+i \pi \right ) \left ({\mathrm e}^{x}-x \right )}{x^{6}}}\) | \(27\) |
Input:
int(((-6+x)*(ln(13/6)+I*Pi)*exp(x)+5*x*(ln(13/6)+I*Pi))*exp(((ln(13/6)+I*P i)*exp(x)-x*(ln(13/6)+I*Pi))/x^6)/x^7,x,method=_RETURNVERBOSE)
Output:
exp((ln(13/6)+I*Pi)/x^6*(exp(x)-x))
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )-x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (e^x (-6+x) \left (i \pi +\log \left (\frac {13}{6}\right )\right )+5 x \left (i \pi +\log \left (\frac {13}{6}\right )\right )\right )}{x^7} \, dx=e^{\left (-\frac {i \, \pi }{x^{5}} + \frac {i \, \pi e^{x}}{x^{6}} - \frac {\log \left (\frac {13}{6}\right )}{x^{5}} + \frac {e^{x} \log \left (\frac {13}{6}\right )}{x^{6}}\right )} \] Input:
integrate(((-6+x)*(log(13/6)+I*pi)*exp(x)+5*x*(log(13/6)+I*pi))*exp(((log( 13/6)+I*pi)*exp(x)-x*(log(13/6)+I*pi))/x^6)/x^7,x, algorithm="fricas")
Output:
e^(-I*pi/x^5 + I*pi*e^x/x^6 - log(13/6)/x^5 + e^x*log(13/6)/x^6)
Timed out. \[ \int \frac {e^{\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )-x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (e^x (-6+x) \left (i \pi +\log \left (\frac {13}{6}\right )\right )+5 x \left (i \pi +\log \left (\frac {13}{6}\right )\right )\right )}{x^7} \, dx=\text {Timed out} \] Input:
integrate(((-6+x)*(ln(13/6)+I*pi)*exp(x)+5*x*(ln(13/6)+I*pi))*exp(((ln(13/ 6)+I*pi)*exp(x)-x*(ln(13/6)+I*pi))/x**6)/x**7,x)
Output:
Timed out
Time = 0.35 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.65 \[ \int \frac {e^{\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )-x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (e^x (-6+x) \left (i \pi +\log \left (\frac {13}{6}\right )\right )+5 x \left (i \pi +\log \left (\frac {13}{6}\right )\right )\right )}{x^7} \, dx=e^{\left (-\frac {i \, \pi }{x^{5}} + \frac {i \, \pi e^{x}}{x^{6}} - \frac {\log \left (13\right )}{x^{5}} + \frac {e^{x} \log \left (13\right )}{x^{6}} + \frac {\log \left (3\right )}{x^{5}} - \frac {e^{x} \log \left (3\right )}{x^{6}} + \frac {\log \left (2\right )}{x^{5}} - \frac {e^{x} \log \left (2\right )}{x^{6}}\right )} \] Input:
integrate(((-6+x)*(log(13/6)+I*pi)*exp(x)+5*x*(log(13/6)+I*pi))*exp(((log( 13/6)+I*pi)*exp(x)-x*(log(13/6)+I*pi))/x^6)/x^7,x, algorithm="maxima")
Output:
e^(-I*pi/x^5 + I*pi*e^x/x^6 - log(13)/x^5 + e^x*log(13)/x^6 + log(3)/x^5 - e^x*log(3)/x^6 + log(2)/x^5 - e^x*log(2)/x^6)
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )-x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (e^x (-6+x) \left (i \pi +\log \left (\frac {13}{6}\right )\right )+5 x \left (i \pi +\log \left (\frac {13}{6}\right )\right )\right )}{x^7} \, dx=e^{\left (-\frac {i \, \pi }{x^{5}} + \frac {i \, \pi e^{x}}{x^{6}} - \frac {\log \left (\frac {13}{6}\right )}{x^{5}} + \frac {e^{x} \log \left (\frac {13}{6}\right )}{x^{6}}\right )} \] Input:
integrate(((-6+x)*(log(13/6)+I*pi)*exp(x)+5*x*(log(13/6)+I*pi))*exp(((log( 13/6)+I*pi)*exp(x)-x*(log(13/6)+I*pi))/x^6)/x^7,x, algorithm="giac")
Output:
e^(-I*pi/x^5 + I*pi*e^x/x^6 - log(13/6)/x^5 + e^x*log(13/6)/x^6)
Time = 3.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {e^{\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )-x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (e^x (-6+x) \left (i \pi +\log \left (\frac {13}{6}\right )\right )+5 x \left (i \pi +\log \left (\frac {13}{6}\right )\right )\right )}{x^7} \, dx=\frac {6^{\frac {1}{x^5}}\,{13}^{\frac {{\mathrm {e}}^x}{x^6}}\,{\mathrm {e}}^{\frac {\Pi \,{\mathrm {e}}^x\,1{}\mathrm {i}}{x^6}}\,{\mathrm {e}}^{-\frac {\Pi \,1{}\mathrm {i}}{x^5}}}{6^{\frac {{\mathrm {e}}^x}{x^6}}\,{13}^{\frac {1}{x^5}}} \] Input:
int((exp(-(x*(Pi*1i + log(13/6)) - exp(x)*(Pi*1i + log(13/6)))/x^6)*(5*x*( Pi*1i + log(13/6)) + exp(x)*(Pi*1i + log(13/6))*(x - 6)))/x^7,x)
Output:
(6^(1/x^5)*13^(exp(x)/x^6)*exp((Pi*exp(x)*1i)/x^6)*exp(-(Pi*1i)/x^5))/(6^( exp(x)/x^6)*13^(1/x^5))
\[ \int \frac {e^{\frac {e^x \left (i \pi +\log \left (\frac {13}{6}\right )\right )-x \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \left (e^x (-6+x) \left (i \pi +\log \left (\frac {13}{6}\right )\right )+5 x \left (i \pi +\log \left (\frac {13}{6}\right )\right )\right )}{x^7} \, dx=\int \frac {\left (\left (-6+x \right ) \left (\mathrm {log}\left (\frac {13}{6}\right )+i \pi \right ) {\mathrm e}^{x}+5 x \left (\mathrm {log}\left (\frac {13}{6}\right )+i \pi \right )\right ) {\mathrm e}^{\frac {\left (\mathrm {log}\left (\frac {13}{6}\right )+i \pi \right ) {\mathrm e}^{x}-x \left (\mathrm {log}\left (\frac {13}{6}\right )+i \pi \right )}{x^{6}}}}{x^{7}}d x \] Input:
int(((-6+x)*(log(13/6)+I*Pi)*exp(x)+5*x*(log(13/6)+I*Pi))*exp(((log(13/6)+ I*Pi)*exp(x)-x*(log(13/6)+I*Pi))/x^6)/x^7,x)
Output:
int(((-6+x)*(log(13/6)+I*Pi)*exp(x)+5*x*(log(13/6)+I*Pi))*exp(((log(13/6)+ I*Pi)*exp(x)-x*(log(13/6)+I*Pi))/x^6)/x^7,x)