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

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

________________________________________________________________________________________

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} $$\begin{array}{c}-16\log \left(e^x(e^{e^{e^x+4}}+3)\right)\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\mathrm{Subst}(\int \frac{16(-3-e^{e^{4+x}}-e^{4+e^{4+x}+x}x)}{(3+e^{e^{4+x}})x}dx,x,e^x)\\ & =16\mathrm{Subst}(\int \frac{-3-e^{e^{4+x}}-e^{4+e^{4+x}+x}x}{(3+e^{e^{4+x}})x}dx,x,e^x)\\ & =-16\log \left(e^x(3+e^{e^{4+e^x}})\right)\end{array}\end{array}$$

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 16, normalized size = 0.76 $$\begin{array}{c}-16(x+\log (3+e^{e^{4+e^x}}))\end{array}$$

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)])

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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))

________________________________________________________________________________________

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

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________