3.91.60 \(\int 8 e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))} x^4 (i \pi +\log (4)) \, dx\)

Optimal. Leaf size=19 \[ e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}} \]

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Rubi [A]  time = 0.26, antiderivative size = 24, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 4, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {12, 6715, 2282, 2194} \begin {gather*} e^{2^{\frac {16 x^5}{5}} e^{\frac {8}{5} i \pi x^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

E^(2^((16*x^5)/5)*E^(((8*I)/5)*Pi*x^5))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6715

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=(8 (i \pi +\log (4))) \int \exp \left (e^{\frac {8}{5} x^5 (i \pi +\log (4))}+\frac {8}{5} x^5 (i \pi +\log (4))\right ) x^4 \, dx\\ &=\frac {1}{5} (8 (i \pi +\log (4))) \operatorname {Subst}\left (\int \exp \left (e^{\frac {8}{5} x (i \pi +\log (4))}+\frac {8}{5} x (i \pi +\log (4))\right ) \, dx,x,x^5\right )\\ &=\operatorname {Subst}\left (\int e^x \, dx,x,e^{\frac {8}{5} x^5 (i \pi +\log (4))}\right )\\ &=e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 19, normalized size = 1.00 \begin {gather*} e^{e^{\frac {8}{5} x^5 (i \pi +\log (4))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

E^E^((8*x^5*(I*Pi + Log[4]))/5)

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fricas [A]  time = 0.56, size = 39, normalized size = 2.05 \begin {gather*} \cosh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \relax (2)\right )}\right ) - \sinh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \relax (2)\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="f
ricas")

[Out]

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)))

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giac [A]  time = 0.31, size = 39, normalized size = 2.05 \begin {gather*} \cosh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \relax (2)\right )}\right ) - \sinh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \relax (2)\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="g
iac")

[Out]

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)))

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maple [A]  time = 0.09, size = 17, normalized size = 0.89

method	result	size
derivativedivides	\({\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \relax (2)+i \pi \right )}{5}}}\)	\(17\)
norman	\({\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \relax (2)+i \pi \right )}{5}}}\)	\(17\)
default	\({\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \relax (2)+i \pi \right )}{5}}}\)	\(39\)
risch	\(\frac {{\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \relax (2)+i \pi \right )}{5}}} \pi }{-2 i \ln \relax (2)+\pi }-\frac {2 i {\mathrm e}^{{\mathrm e}^{\frac {8 x^{5} \left (2 \ln \relax (2)+i \pi \right )}{5}}} \ln \relax (2)}{-2 i \ln \relax (2)+\pi }\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

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maxima [A]  time = 0.57, size = 39, normalized size = 2.05 \begin {gather*} \cosh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \relax (2)\right )}\right ) - \sinh \left (-e^{\left (\frac {8}{5} i \, \pi x^{5} + \frac {16}{5} \, x^{5} \log \relax (2)\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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="m
axima")

[Out]

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)))

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mupad [B]  time = 0.60, size = 17, normalized size = 0.89 \begin {gather*} {\mathrm {e}}^{2^{\frac {16\,x^5}{5}}\,{\mathrm {e}}^{\frac {\Pi \,x^5\,8{}\mathrm {i}}{5}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

[Out]

Timed out

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