3.54.44 \(\int \frac {e^{\frac {e^x (i \pi +\log (\frac {13}{6}))-x (i \pi +\log (\frac {13}{6}))}{x^6}} (e^x (-6+x) (i \pi +\log (\frac {13}{6}))+5 x (i \pi +\log (\frac {13}{6})))}{x^7} \, dx\)

Optimal. Leaf size=23 \[ e^{\frac {\left (e^x-x\right ) \left (i \pi +\log \left (\frac {13}{6}\right )\right )}{x^6}} \]

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Rubi [A]  time = 1.14, antiderivative size = 34, normalized size of antiderivative = 1.48, number of steps used = 1, number of rules used = 1, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6706} \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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 + Lo
g[13/6])))/x^7,x]

[Out]

E^((E^x*(I*Pi + Log[13/6]) - x*(I*Pi + Log[13/6]))/x^6)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\exp \left (\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}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 2.25, size = 61, normalized size = 2.65 \begin {gather*} -\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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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*P
i + Log[13/6])))/x^7,x]

[Out]

((-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])

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fricas [A]  time = 0.63, size = 31, normalized size = 1.35 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-6)*(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")

[Out]

e^(-I*pi/x^5 + I*pi*e^x/x^6 - log(13/6)/x^5 + e^x*log(13/6)/x^6)

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giac [A]  time = 0.24, size = 31, normalized size = 1.35 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-6)*(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")

[Out]

e^(-I*pi/x^5 + I*pi*e^x/x^6 - log(13/6)/x^5 + e^x*log(13/6)/x^6)

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maple [A]  time = 0.48, size = 27, normalized size = 1.17

method	result	size
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	\({\mathrm e}^{\frac {\left (i \pi -\ln \relax (3)-\ln \relax (2)+\ln \left (13\right )\right ) \left ({\mathrm e}^{x}-x \right )}{x^{6}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x-6)*(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,method=_RETURNVERBOSE)

[Out]

exp(((ln(13/6)+I*Pi)*exp(x)-x*(ln(13/6)+I*Pi))/x^6)

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maxima [A]  time = 0.77, size = 61, normalized size = 2.65 \begin {gather*} 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 \relax (3)}{x^{5}} - \frac {e^{x} \log \relax (3)}{x^{6}} + \frac {\log \relax (2)}{x^{5}} - \frac {e^{x} \log \relax (2)}{x^{6}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-6)*(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")

[Out]

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*l
og(2)/x^6)

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mupad [B]  time = 3.99, size = 49, normalized size = 2.13 \begin {gather*} \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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

(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))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x-6)*(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)

[Out]

Timed out

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