3.1.60 \(\int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx\) [60]

3.1.60.1 Optimal result

Integrand size = 48, antiderivative size = 19 \begin {dmath*} \int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx=e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}} \end {dmath*}

output

exp(exp(8/5*x^5*(2*ln(2)+I*Pi)))
 

3.1.60.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \begin {dmath*} \int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx=e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}} \end {dmath*}

input

Integrate[8*E^(E^((8*x^5*(I*Pi + Log[4]))/5) + (8*x^5*(I*Pi + Log[4]))/5)* 
x^4*(I*Pi + Log[4]),x]
 

output

E^E^((8*x^5*(I*Pi + Log[4]))/5)
 

3.1.60.3 Rubi [F]

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.

\(\displaystyle \int 8 x^4 (\log (4)+i \pi ) \exp \left (\frac {8}{5} x^5 (\log (4)+i \pi )+e^{\frac {8}{5} x^5 (\log (4)+i \pi )}\right ) \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle 8 (\log (4)+i \pi ) \int \exp \left (\frac {8}{5} (i \pi +\log (4)) x^5+2^{\frac {16 x^5}{5}} e^{\frac {8}{5} i \pi x^5}\right ) x^4dx\)

\(\Big \downarrow \) 7266

\(\displaystyle \frac {8}{5} (\log (4)+i \pi ) \int \exp \left (\frac {8}{5} (i \pi +\log (4)) x^5+2^{\frac {16 x^5}{5}} e^{\frac {8}{5} i \pi x^5}\right )dx^5\)

\(\Big \downarrow \) 7281

\(\displaystyle (\log (4)+i \pi ) \int \exp \left (\frac {8}{5} (i \pi +\log (4)) x^5+2^{\frac {16 x^5}{5}} e^{\frac {8}{5} i \pi x^5}\right )d\frac {8 x^5}{5}\)

\(\Big \downarrow \) 7299

\(\displaystyle (\log (4)+i \pi ) \int \exp \left (\frac {8}{5} (i \pi +\log (4)) x^5+2^{\frac {16 x^5}{5}} e^{\frac {8}{5} i \pi x^5}\right )d\frac {8 x^5}{5}\)

input

Int[8*E^(E^((8*x^5*(I*Pi + Log[4]))/5) + (8*x^5*(I*Pi + Log[4]))/5)*x^4*(I 
*Pi + Log[4]),x]
 

output

$Aborted
 

3.1.60.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 7266

Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1)   Subst[Int[SubstFor[x^(m 
+ 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function 
OfQ[x^(m + 1), u, x]
 

rule 7281

Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] 
 Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /;  !FalseQ[lst]]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

3.1.60.4 Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\({\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \left (2\right )+i \pi \right )}{5}}}\)	\(17\)
norman	\({\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \left (2\right )+i \pi \right )}{5}}}\)	\(17\)
default	\({\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \left (2\right )+i \pi \right )}{5}}}\)	\(39\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \left (2\right )+i \pi \right )}{5}}}\)	\(39\)
risch	\(\frac {{\mathrm e}^{2^{\frac {16 x^{5}}{5}} {\mathrm e}^{\frac {8 i \pi \,x^{5}}{5}}} \pi }{-2 i \ln \left (2\right )+\pi }-\frac {2 i {\mathrm e}^{2^{\frac {16 x^{5}}{5}} {\mathrm e}^{\frac {8 i \pi \,x^{5}}{5}}} \ln \left (2\right )}{-2 i \ln \left (2\right )+\pi }\)	\(61\)

input

int(8*x^4*(2*ln(2)+I*Pi)*exp(8/5*x^5*(2*ln(2)+I*Pi))*exp(exp(8/5*x^5*(2*ln 
(2)+I*Pi))),x,method=_RETURNVERBOSE)
 

output

exp(exp(8/5*x^5*(2*ln(2)+I*Pi)))
 

3.1.60.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \begin {dmath*} \int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx=\cosh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \left (2\right )\right )}\right ) - \sinh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \left (2\right )\right )}\right ) \end {dmath*}

input

integrate(8*x^4*(2*log(2)+I*pi)*exp(8/5*x^5*(2*log(2)+I*pi))*exp(exp(8/5*x 
^5*(2*log(2)+I*pi))),x, algorithm=\
 

output

cosh(-e^(8/5*I*pi*x^5 + 16/5*x^5*log(2))) - sinh(-e^(8/5*I*pi*x^5 + 16/5*x 
^5*log(2)))
 

3.1.60.6 Sympy [A] (verification not implemented)

Time = 82.59 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \begin {dmath*} \int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx=e^{e^{\frac {8 x^{5} \cdot \left (2 \log {\left (2 \right )} + i \pi \right )}{5}}} \end {dmath*}

input

integrate(8*x**4*(2*ln(2)+I*pi)*exp(8/5*x**5*(2*ln(2)+I*pi))*exp(exp(8/5*x 
**5*(2*ln(2)+I*pi))),x)
 

output

exp(exp(8*x**5*(2*log(2) + I*pi)/5))
 

3.1.60.7 Maxima [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \begin {dmath*} \int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx=\cosh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \left (2\right )\right )}\right ) - \sinh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \left (2\right )\right )}\right ) \end {dmath*}

input

integrate(8*x^4*(2*log(2)+I*pi)*exp(8/5*x^5*(2*log(2)+I*pi))*exp(exp(8/5*x 
^5*(2*log(2)+I*pi))),x, algorithm=\
 

output

cosh(-e^(8/5*I*pi*x^5 + 16/5*x^5*log(2))) - sinh(-e^(8/5*I*pi*x^5 + 16/5*x 
^5*log(2)))
 

3.1.60.8 Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \begin {dmath*} \int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx=\cosh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \left (2\right )\right )}\right ) - \sinh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \left (2\right )\right )}\right ) \end {dmath*}

input

integrate(8*x^4*(2*log(2)+I*pi)*exp(8/5*x^5*(2*log(2)+I*pi))*exp(exp(8/5*x 
^5*(2*log(2)+I*pi))),x, algorithm=\
 

output

cosh(-e^(8/5*I*pi*x^5 + 16/5*x^5*log(2))) - sinh(-e^(8/5*I*pi*x^5 + 16/5*x 
^5*log(2)))
 

3.1.60.9 Mupad [B] (verification not implemented)

Time = 13.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \begin {dmath*} \int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx={\mathrm {e}}^{2^{\frac {16\,x^5}{5}}\,{\mathrm {e}}^{\frac {\Pi \,x^5\,8{}\mathrm {i}}{5}}} \end {dmath*}

input

int(8*x^4*exp((8*x^5*(Pi*1i + 2*log(2)))/5)*exp(exp((8*x^5*(Pi*1i + 2*log( 
2)))/5))*(Pi*1i + 2*log(2)),x)
 

output

exp(2^((16*x^5)/5)*exp((Pi*x^5*8i)/5))