3.41.34 \(\int \frac {-48+e^{e^{4+e^x}} (-16-16 e^{4+e^x+x})}{3+e^{e^{4+e^x}}} \, dx\)

Optimal. Leaf size=21 \[ 16 \left (15-x-\log \left (3+e^{e^{4+e^x}}\right )\right ) \]

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Rubi [A]  time = 0.22, antiderivative size = 18, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2282, 12, 6685} \begin {gather*} -16 \log \left (e^x \left (e^{e^{e^x+4}}+3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-48 + E^E^(4 + E^x)*(-16 - 16*E^(4 + E^x + x)))/(3 + E^E^(4 + E^x)),x]

[Out]

-16*Log[E^x*(3 + E^E^(4 + E^x))]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 6685

Int[(u_)/((w_)*(y_)), x_Symbol] :> With[{q = DerivativeDivides[y*w, u, x]}, Simp[q*Log[RemoveContent[y*w, x]],
x] /;  !FalseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {16 \left (-3-e^{e^{4+x}}-e^{4+e^{4+x}+x} x\right )}{\left (3+e^{e^{4+x}}\right ) x} \, dx,x,e^x\right )\\ &=16 \operatorname {Subst}\left (\int \frac {-3-e^{e^{4+x}}-e^{4+e^{4+x}+x} x}{\left (3+e^{e^{4+x}}\right ) x} \, dx,x,e^x\right )\\ &=-16 \log \left (e^x \left (3+e^{e^{4+e^x}}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 16, normalized size = 0.76 \begin {gather*} -16 \left (x+\log \left (3+e^{e^{4+e^x}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-48 + E^E^(4 + E^x)*(-16 - 16*E^(4 + E^x + x)))/(3 + E^E^(4 + E^x)),x]

[Out]

-16*(x + Log[3 + E^E^(4 + E^x)])

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fricas [A]  time = 0.72, size = 15, normalized size = 0.71 \begin {gather*} -16 \, x - 16 \, \log \left (e^{\left (e^{\left (e^{x} + 4\right )}\right )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*exp(4)*exp(x)*exp(exp(x))-16)*exp(exp(4)*exp(exp(x)))-48)/(exp(exp(4)*exp(exp(x)))+3),x, algor
ithm="fricas")

[Out]

-16*x - 16*log(e^(e^(e^x + 4)) + 3)

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giac [A]  time = 0.21, size = 29, normalized size = 1.38 \begin {gather*} -16 \, x + 16 \, e^{x} - 16 \, \log \left (e^{\left (e^{x} + e^{\left (e^{x} + 4\right )} + 4\right )} + 3 \, e^{\left (e^{x} + 4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*exp(4)*exp(x)*exp(exp(x))-16)*exp(exp(4)*exp(exp(x)))-48)/(exp(exp(4)*exp(exp(x)))+3),x, algor
ithm="giac")

[Out]

-16*x + 16*e^x - 16*log(e^(e^x + e^(e^x + 4) + 4) + 3*e^(e^x + 4))

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maple [A]  time = 0.14, size = 16, normalized size = 0.76

method	result	size
risch	\(-16 x -16 \ln \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+4}}+3\right )\)	\(16\)
norman	\(-16 x -16 \ln \left ({\mathrm e}^{{\mathrm e}^{4} {\mathrm e}^{{\mathrm e}^{x}}}+3\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*exp(4)*exp(x)*exp(exp(x))-16)*exp(exp(4)*exp(exp(x)))-48)/(exp(exp(4)*exp(exp(x)))+3),x,method=_RETU
RNVERBOSE)

[Out]

-16*x-16*ln(exp(exp(exp(x)+4))+3)

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maxima [A]  time = 0.39, size = 15, normalized size = 0.71 \begin {gather*} -16 \, x - 16 \, \log \left (e^{\left (e^{\left (e^{x} + 4\right )}\right )} + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*exp(4)*exp(x)*exp(exp(x))-16)*exp(exp(4)*exp(exp(x)))-48)/(exp(exp(4)*exp(exp(x)))+3),x, algor
ithm="maxima")

[Out]

-16*x - 16*log(e^(e^(e^x + 4)) + 3)

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mupad [B]  time = 3.15, size = 16, normalized size = 0.76 \begin {gather*} -16\,x-16\,\ln \left ({\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^4}+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(exp(x))*exp(4))*(16*exp(exp(x))*exp(4)*exp(x) + 16) + 48)/(exp(exp(exp(x))*exp(4)) + 3),x)

[Out]

- 16*x - 16*log(exp(exp(exp(x))*exp(4)) + 3)

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sympy [A]  time = 0.16, size = 19, normalized size = 0.90 \begin {gather*} - 16 x - 16 \log {\left (e^{e^{4} e^{e^{x}}} + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*exp(4)*exp(x)*exp(exp(x))-16)*exp(exp(4)*exp(exp(x)))-48)/(exp(exp(4)*exp(exp(x)))+3),x)

[Out]

-16*x - 16*log(exp(exp(4)*exp(exp(x))) + 3)

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