Integrand size = 57, antiderivative size = 31 \[ \int \frac {-2 x+2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )-e^{\frac {2+x}{2}} x \left (i \pi +\log \left (\frac {21}{5}\right )\right )}{2 x \left (i \pi +\log \left (\frac {21}{5}\right )\right )} \, dx=-e^{\frac {1}{4} (4+2 x)}-\frac {x}{i \pi +\log \left (\frac {21}{5}\right )}+\log (x) \]
Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {-2 x+2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )-e^{\frac {2+x}{2}} x \left (i \pi +\log \left (\frac {21}{5}\right )\right )}{2 x \left (i \pi +\log \left (\frac {21}{5}\right )\right )} \, dx=-e^{1+\frac {x}{2}}-\frac {x}{i \pi +\log \left (\frac {21}{5}\right )}+\log (x) \]
Integrate[(-2*x + 2*(I*Pi + Log[21/5]) - E^((2 + x)/2)*x*(I*Pi + Log[21/5] ))/(2*x*(I*Pi + Log[21/5])),x]
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {27, 2010, 2009}
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.
\(\displaystyle \int \frac {-2 x-e^{\frac {x+2}{2}} x \left (\log \left (\frac {21}{5}\right )+i \pi \right )+2 \left (\log \left (\frac {21}{5}\right )+i \pi \right )}{2 x \left (\log \left (\frac {21}{5}\right )+i \pi \right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {-e^{\frac {x+2}{2}} \left (i \pi +\log \left (\frac {21}{5}\right )\right ) x-2 x+\log \left (\frac {441}{25}\right )+2 i \pi }{x}dx}{2 \left (\log \left (\frac {21}{5}\right )+i \pi \right )}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {\int \left (\frac {i \left (2 i x-i \log \left (\frac {441}{25}\right )+2 \pi \right )}{x}-i e^{\frac {x}{2}+1} \left (\pi -i \log \left (\frac {21}{5}\right )\right )\right )dx}{2 \left (\log \left (\frac {21}{5}\right )+i \pi \right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-2 x+\left (\log \left (\frac {441}{25}\right )+2 i \pi \right ) \log (x)-2 e^{\frac {x}{2}+1} \left (\log \left (\frac {21}{5}\right )+i \pi \right )}{2 \left (\log \left (\frac {21}{5}\right )+i \pi \right )}\) |
Int[(-2*x + 2*(I*Pi + Log[21/5]) - E^((2 + x)/2)*x*(I*Pi + Log[21/5]))/(2* x*(I*Pi + Log[21/5])),x]
(-2*x - 2*E^(1 + x/2)*(I*Pi + Log[21/5]) + ((2*I)*Pi + Log[441/25])*Log[x] )/(2*(I*Pi + Log[21/5]))
3.26.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10
method | result | size |
parts | \(\frac {-x +\left (\ln \left (\frac {21}{5}\right )+i \pi \right ) \ln \left (x \right )}{\ln \left (\frac {21}{5}\right )+i \pi }-{\mathrm e}^{1+\frac {x}{2}}\) | \(34\) |
norman | \(\frac {\left (i \pi -\ln \left (21\right )+\ln \left (5\right )\right ) x}{\ln \left (21\right )^{2}-2 \ln \left (21\right ) \ln \left (5\right )+\ln \left (5\right )^{2}+\pi ^{2}}-{\mathrm e}^{1+\frac {x}{2}}+\ln \left (x \right )\) | \(45\) |
parallelrisch | \(\frac {2 i \ln \left (x \right ) \pi -2 i \pi \,{\mathrm e}^{1+\frac {x}{2}}+2 \ln \left (x \right ) \ln \left (\frac {21}{5}\right )-2 \ln \left (\frac {21}{5}\right ) {\mathrm e}^{1+\frac {x}{2}}-2 x}{2 \ln \left (\frac {21}{5}\right )+2 i \pi }\) | \(48\) |
risch | \(-\frac {x}{\ln \left (3\right )+\ln \left (7\right )-\ln \left (5\right )+i \pi }+\ln \left (x \right )-\frac {i {\mathrm e}^{1+\frac {x}{2}} \pi }{\ln \left (3\right )+\ln \left (7\right )-\ln \left (5\right )+i \pi }-\frac {{\mathrm e}^{1+\frac {x}{2}} \ln \left (3\right )}{\ln \left (3\right )+\ln \left (7\right )-\ln \left (5\right )+i \pi }-\frac {{\mathrm e}^{1+\frac {x}{2}} \ln \left (7\right )}{\ln \left (3\right )+\ln \left (7\right )-\ln \left (5\right )+i \pi }+\frac {{\mathrm e}^{1+\frac {x}{2}} \ln \left (5\right )}{\ln \left (3\right )+\ln \left (7\right )-\ln \left (5\right )+i \pi }\) | \(121\) |
derivativedivides | \(\frac {2 i \pi \ln \left (\frac {x}{2}\right )-2 i \pi \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )-2 i \pi \left ({\mathrm e}^{1+\frac {x}{2}}-{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )\right )-4-2 x -2 \ln \left (5\right ) \ln \left (\frac {x}{2}\right )+2 \ln \left (21\right ) \ln \left (\frac {x}{2}\right )+2 \ln \left (5\right ) {\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )-2 \ln \left (21\right ) {\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )+2 \ln \left (5\right ) \left ({\mathrm e}^{1+\frac {x}{2}}-{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )\right )-2 \ln \left (21\right ) \left ({\mathrm e}^{1+\frac {x}{2}}-{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )\right )}{2 \ln \left (\frac {21}{5}\right )+2 i \pi }\) | \(134\) |
default | \(\frac {2 i \pi \ln \left (\frac {x}{2}\right )-2 i \pi \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )-2 i \pi \left ({\mathrm e}^{1+\frac {x}{2}}-{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )\right )-4-2 x -2 \ln \left (5\right ) \ln \left (\frac {x}{2}\right )+2 \ln \left (21\right ) \ln \left (\frac {x}{2}\right )+2 \ln \left (5\right ) {\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )-2 \ln \left (21\right ) {\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )+2 \ln \left (5\right ) \left ({\mathrm e}^{1+\frac {x}{2}}-{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )\right )-2 \ln \left (21\right ) \left ({\mathrm e}^{1+\frac {x}{2}}-{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {x}{2}\right )\right )}{2 \ln \left (\frac {21}{5}\right )+2 i \pi }\) | \(134\) |
int(1/2*(-x*(ln(21/5)+I*Pi)*exp(1+1/2*x)+2*ln(21/5)+2*I*Pi-2*x)/x/(ln(21/5 )+I*Pi),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {-2 x+2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )-e^{\frac {2+x}{2}} x \left (i \pi +\log \left (\frac {21}{5}\right )\right )}{2 x \left (i \pi +\log \left (\frac {21}{5}\right )\right )} \, dx=\frac {{\left (-i \, \pi - \log \left (\frac {21}{5}\right )\right )} e^{\left (\frac {1}{2} \, x + 1\right )} + {\left (i \, \pi + \log \left (\frac {21}{5}\right )\right )} \log \left (x\right ) - x}{i \, \pi + \log \left (\frac {21}{5}\right )} \]
integrate(1/2*(-x*(log(21/5)+I*pi)*exp(1+1/2*x)+2*log(21/5)+2*I*pi-2*x)/x/ (log(21/5)+I*pi),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-2 x+2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )-e^{\frac {2+x}{2}} x \left (i \pi +\log \left (\frac {21}{5}\right )\right )}{2 x \left (i \pi +\log \left (\frac {21}{5}\right )\right )} \, dx=\frac {- x - \left (- \log {\left (21 \right )} + \log {\left (5 \right )} - i \pi \right ) \log {\left (x \right )}}{- \log {\left (5 \right )} + \log {\left (21 \right )} + i \pi } - e e^{\frac {x}{2}} \]
Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-2 x+2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )-e^{\frac {2+x}{2}} x \left (i \pi +\log \left (\frac {21}{5}\right )\right )}{2 x \left (i \pi +\log \left (\frac {21}{5}\right )\right )} \, dx=\frac {-i \, \pi e^{\left (\frac {1}{2} \, x + 1\right )} - e^{\left (\frac {1}{2} \, x + 1\right )} \log \left (\frac {21}{5}\right ) + i \, \pi \log \left (x\right ) + \log \left (\frac {21}{5}\right ) \log \left (x\right ) - x}{i \, \pi + \log \left (\frac {21}{5}\right )} \]
integrate(1/2*(-x*(log(21/5)+I*pi)*exp(1+1/2*x)+2*log(21/5)+2*I*pi-2*x)/x/ (log(21/5)+I*pi),x, algorithm=\
(-I*pi*e^(1/2*x + 1) - e^(1/2*x + 1)*log(21/5) + I*pi*log(x) + log(21/5)*l og(x) - x)/(I*pi + log(21/5))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {-2 x+2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )-e^{\frac {2+x}{2}} x \left (i \pi +\log \left (\frac {21}{5}\right )\right )}{2 x \left (i \pi +\log \left (\frac {21}{5}\right )\right )} \, dx=-\frac {i \, \pi e^{\left (\frac {1}{2} \, x + 1\right )} + e^{\left (\frac {1}{2} \, x + 1\right )} \log \left (\frac {21}{5}\right ) - i \, \pi \log \left (\frac {1}{2} \, x\right ) - \log \left (\frac {21}{5}\right ) \log \left (\frac {1}{2} \, x\right ) + x}{i \, \pi + \log \left (\frac {21}{5}\right )} \]
integrate(1/2*(-x*(log(21/5)+I*pi)*exp(1+1/2*x)+2*log(21/5)+2*I*pi-2*x)/x/ (log(21/5)+I*pi),x, algorithm=\
-(I*pi*e^(1/2*x + 1) + e^(1/2*x + 1)*log(21/5) - I*pi*log(1/2*x) - log(21/ 5)*log(1/2*x) + x)/(I*pi + log(21/5))
Time = 11.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {-2 x+2 \left (i \pi +\log \left (\frac {21}{5}\right )\right )-e^{\frac {2+x}{2}} x \left (i \pi +\log \left (\frac {21}{5}\right )\right )}{2 x \left (i \pi +\log \left (\frac {21}{5}\right )\right )} \, dx=\ln \left (x\right )-\frac {{\mathrm {e}}^{\frac {x}{2}+1}\,\left (2\,\Pi ^2+2\,{\ln \left (\frac {21}{5}\right )}^2\right )-x\,\left (-2\,\ln \left (\frac {21}{5}\right )+\Pi \,2{}\mathrm {i}\right )}{2\,\Pi ^2+2\,{\ln \left (\frac {21}{5}\right )}^2} \]
int((Pi*1i - x + log(21/5) - (x*exp(x/2 + 1)*(Pi*1i + log(21/5)))/2)/(x*(P i*1i + log(21/5))),x)