3.83.93 \(\int \frac {12+e^{e^x} (4+e^x x)}{3 x+e^{e^x} x} \, dx\)

Optimal. Leaf size=20 \[ 9-e^4+\log \left (2 \left (3+e^{e^x}\right ) x^4\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 13, normalized size of antiderivative = 0.65, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {6742, 2282, 2246, 31} \begin {gather*} \log \left (e^{e^x}+3\right )+4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12 + E^E^x*(4 + E^x*x))/(3*x + E^E^x*x),x]

[Out]

Log[3 + E^E^x] + 4*Log[x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, 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 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{e^x+x}}{3+e^{e^x}}+\frac {4}{x}\right ) \, dx\\ &=4 \log (x)+\int \frac {e^{e^x+x}}{3+e^{e^x}} \, dx\\ &=4 \log (x)+\operatorname {Subst}\left (\int \frac {e^x}{3+e^x} \, dx,x,e^x\right )\\ &=4 \log (x)+\operatorname {Subst}\left (\int \frac {1}{3+x} \, dx,x,e^{e^x}\right )\\ &=\log \left (3+e^{e^x}\right )+4 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 13, normalized size = 0.65 \begin {gather*} \log \left (3+e^{e^x}\right )+4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 + E^E^x*(4 + E^x*x))/(3*x + E^E^x*x),x]

[Out]

Log[3 + E^E^x] + 4*Log[x]

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fricas [A]  time = 0.65, size = 11, normalized size = 0.55 \begin {gather*} 4 \, \log \relax (x) + \log \left (e^{\left (e^{x}\right )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x+4)*exp(exp(x))+12)/(x*exp(exp(x))+3*x),x, algorithm="fricas")

[Out]

4*log(x) + log(e^(e^x) + 3)

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giac [A]  time = 0.14, size = 19, normalized size = 0.95 \begin {gather*} -x + 4 \, \log \relax (x) + \log \left (e^{\left (x + e^{x}\right )} + 3 \, e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x+4)*exp(exp(x))+12)/(x*exp(exp(x))+3*x),x, algorithm="giac")

[Out]

-x + 4*log(x) + log(e^(x + e^x) + 3*e^x)

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maple [A]  time = 0.04, size = 12, normalized size = 0.60

method	result	size
norman	\(4 \ln \relax (x )+\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+3\right )\)	\(12\)
risch	\(4 \ln \relax (x )+\ln \left ({\mathrm e}^{{\mathrm e}^{x}}+3\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*x+4)*exp(exp(x))+12)/(x*exp(exp(x))+3*x),x,method=_RETURNVERBOSE)

[Out]

4*ln(x)+ln(exp(exp(x))+3)

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maxima [A]  time = 0.40, size = 11, normalized size = 0.55 \begin {gather*} 4 \, \log \relax (x) + \log \left (e^{\left (e^{x}\right )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x+4)*exp(exp(x))+12)/(x*exp(exp(x))+3*x),x, algorithm="maxima")

[Out]

4*log(x) + log(e^(e^x) + 3)

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mupad [B]  time = 0.06, size = 11, normalized size = 0.55 \begin {gather*} \ln \left ({\mathrm {e}}^{{\mathrm {e}}^x}+3\right )+4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x))*(x*exp(x) + 4) + 12)/(3*x + x*exp(exp(x))),x)

[Out]

log(exp(exp(x)) + 3) + 4*log(x)

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sympy [A]  time = 0.16, size = 12, normalized size = 0.60 \begin {gather*} 4 \log {\relax (x )} + \log {\left (e^{e^{x}} + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)*x+4)*exp(exp(x))+12)/(x*exp(exp(x))+3*x),x)

[Out]

4*log(x) + log(exp(exp(x)) + 3)

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