3.21.26 \(\int (2+e^{e^{e^{4+x}}+x} (1+e^{4+e^{4+x}+x})-\log (4)) \, dx\)

Optimal. Leaf size=20 \[ 5+e^{e^{e^{4+x}}+x}-x (-2+\log (4)) \]

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Rubi [A]  time = 0.08, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2282, 2288} \begin {gather*} e^{x+e^{e^{x+4}}}+x (2-\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + E^(E^E^(4 + x) + x)*(1 + E^(4 + E^(4 + x) + x)) - Log[4],x]

[Out]

E^(E^E^(4 + x) + x) + x*(2 - Log[4])

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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x (2-\log (4))+\int e^{e^{e^{4+x}}+x} \left (1+e^{4+e^{4+x}+x}\right ) \, dx\\ &=x (2-\log (4))+\operatorname {Subst}\left (\int e^{e^{e^4 x}} \left (1+e^{4+e^4 x} x\right ) \, dx,x,e^x\right )\\ &=e^{e^{e^{4+x}}+x}+x (2-\log (4))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 20, normalized size = 1.00 \begin {gather*} e^{e^{e^{4+x}}+x}+2 x-x \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + E^(E^E^(4 + x) + x)*(1 + E^(4 + E^(4 + x) + x)) - Log[4],x]

[Out]

E^(E^E^(4 + x) + x) + 2*x - x*Log[4]

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fricas [A]  time = 0.88, size = 32, normalized size = 1.60 \begin {gather*} -2 \, x \log \relax (2) + 2 \, x + e^{\left ({\left (x e^{\left (x + 4\right )} + e^{\left (x + e^{\left (x + 4\right )} + 4\right )}\right )} e^{\left (-x - 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(2) + 2*x + e^((x*e^(x + 4) + e^(x + e^(x + 4) + 4))*e^(-x - 4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (e^{\left (x + e^{\left (x + 4\right )} + 4\right )} + 1\right )} e^{\left (x + e^{\left (e^{\left (x + 4\right )}\right )}\right )} - 2 \, \log \relax (2) + 2\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e^(x + e^(x + 4) + 4) + 1)*e^(x + e^(e^(x + 4))) - 2*log(2) + 2, x)

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maple [A]  time = 0.04, size = 18, normalized size = 0.90

method	result	size
default	\(2 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4+x}}+x}-2 x \ln \relax (2)\)	\(18\)
norman	\(\left (2-2 \ln \relax (2)\right ) x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4+x}}+x}\)	\(18\)
risch	\(2 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4+x}}+x}-2 x \ln \relax (2)\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x+exp(exp(exp(4+x))+x)-2*x*ln(2)

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maxima [A]  time = 0.41, size = 17, normalized size = 0.85 \begin {gather*} -2 \, x \log \relax (2) + 2 \, x + e^{\left (x + e^{\left (e^{\left (x + 4\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(2) + 2*x + e^(x + e^(e^(x + 4)))

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mupad [B]  time = 1.16, size = 17, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{x+{\mathrm {e}}^{{\mathrm {e}}^4\,{\mathrm {e}}^x}}-x\,\left (\ln \relax (4)-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x + exp(exp(4)*exp(x))) - x*(log(4) - 2)

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sympy [A]  time = 0.20, size = 17, normalized size = 0.85 \begin {gather*} x \left (2 - 2 \log {\relax (2 )}\right ) + e^{x + e^{e^{x + 4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(2 - 2*log(2)) + exp(x + exp(exp(x + 4)))

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