3.49.28 \(\int \frac {4 e^{e^{x+x \log (x)}+i \pi +x} (1+e^{x+x \log (x)} (2+\log (x)))}{5+5 \log (3)} \, dx\)

Optimal. Leaf size=29 \[ -\frac {4 e^{e^{x+x \log (x)}+i \pi +x}}{5 (-1-\log (3))} \]

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Rubi [A]  time = 0.11, antiderivative size = 24, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 6706} \begin {gather*} \frac {4 e^{e^x x^x+x+i \pi }}{5+\log (243)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^(E^(x + x*Log[x]) + I*Pi + x)*(1 + E^(x + x*Log[x])*(2 + Log[x])))/(5 + 5*Log[3]),x]

[Out]

(4*E^(I*Pi + x + E^x*x^x))/(5 + Log[243])

Rule 12

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

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} &=\frac {4 \int e^{e^{x+x \log (x)}+i \pi +x} \left (1+e^{x+x \log (x)} (2+\log (x))\right ) \, dx}{5+\log (243)}\\ &=\frac {4 e^{i \pi +x+e^x x^x}}{5+\log (243)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 19, normalized size = 0.66 \begin {gather*} -\frac {4 e^{x+e^x x^x}}{5+\log (243)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^(E^(x + x*Log[x]) + I*Pi + x)*(1 + E^(x + x*Log[x])*(2 + Log[x])))/(5 + 5*Log[3]),x]

[Out]

(-4*E^(x + E^x*x^x))/(5 + Log[243])

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fricas [A]  time = 0.85, size = 19, normalized size = 0.66 \begin {gather*} e^{\left (x + e^{\left (x \log \relax (x) + x\right )} + \log \left (-\frac {4}{5 \, {\left (\log \relax (3) + 1\right )}}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)+2)*exp(x*log(x)+x)+1)*exp(exp(x*log(x)+x)+log(-4/(5*log(3)+5))+x),x, algorithm="fricas")

[Out]

e^(x + e^(x*log(x) + x) + log(-4/5/(log(3) + 1)))

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giac [A]  time = 0.16, size = 19, normalized size = 0.66 \begin {gather*} e^{\left (x + e^{\left (x \log \relax (x) + x\right )} + \log \left (-\frac {4}{5 \, {\left (\log \relax (3) + 1\right )}}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)+2)*exp(x*log(x)+x)+1)*exp(exp(x*log(x)+x)+log(-4/(5*log(3)+5))+x),x, algorithm="giac")

[Out]

e^(x + e^(x*log(x) + x) + log(-4/5/(log(3) + 1)))

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maple [A]  time = 0.04, size = 18, normalized size = 0.62

method	result	size
risch	\(-\frac {4 \,{\mathrm e}^{x^{x} {\mathrm e}^{x}+x}}{5 \left (\ln \relax (3)+1\right )}\)	\(18\)
derivativedivides	\({\mathrm e}^{{\mathrm e}^{x \ln \relax (x )+x}+\ln \left (-\frac {4}{5 \ln \relax (3)+5}\right )+x}\)	\(22\)
default	\({\mathrm e}^{{\mathrm e}^{x \ln \relax (x )+x}+\ln \left (-\frac {4}{5 \ln \relax (3)+5}\right )+x}\)	\(22\)
norman	\({\mathrm e}^{{\mathrm e}^{x \ln \relax (x )+x}+\ln \left (-\frac {4}{5 \ln \relax (3)+5}\right )+x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(x)+2)*exp(x*ln(x)+x)+1)*exp(exp(x*ln(x)+x)+ln(-4/(5*ln(3)+5))+x),x,method=_RETURNVERBOSE)

[Out]

-4/5/(ln(3)+1)*exp(x^x*exp(x)+x)

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maxima [A]  time = 0.35, size = 18, normalized size = 0.62 \begin {gather*} -\frac {4 \, e^{\left (x + e^{\left (x \log \relax (x) + x\right )}\right )}}{5 \, {\left (\log \relax (3) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)+2)*exp(x*log(x)+x)+1)*exp(exp(x*log(x)+x)+log(-4/(5*log(3)+5))+x),x, algorithm="maxima")

[Out]

-4/5*e^(x + e^(x*log(x) + x))/(log(3) + 1)

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mupad [B]  time = 3.37, size = 17, normalized size = 0.59 \begin {gather*} -\frac {4\,{\mathrm {e}}^x\,{\mathrm {e}}^{x^x\,{\mathrm {e}}^x}}{\ln \left (243\right )+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x + log(-4/(5*log(3) + 5)) + exp(x + x*log(x)))*(exp(x + x*log(x))*(log(x) + 2) + 1),x)

[Out]

-(4*exp(x)*exp(x^x*exp(x)))/(log(243) + 5)

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sympy [A]  time = 0.54, size = 20, normalized size = 0.69 \begin {gather*} - \frac {4 e^{x + e^{x \log {\relax (x )} + x}}}{5 + 5 \log {\relax (3 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((ln(x)+2)*exp(x*ln(x)+x)+1)*exp(exp(x*ln(x)+x)+ln(-4/(5*ln(3)+5))+x),x)

[Out]

-4*exp(x + exp(x*log(x) + x))/(5 + 5*log(3))

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