\(\int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx\) [6814]

Optimal result

Integrand size = 30, antiderivative size = 17 \[ \int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx=\log \left (5-e^{-x}-e^{8 x}\right ) \]

[Out]

ln(5-exp(-x)-exp(8*x))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2320, 6874, 1601} \[ \int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx=\log \left (-5 e^x+e^{9 x}+1\right )-x \]

[In]

Int[(-E^(-x) + 8*E^(8*x))/(-5 + E^(-x) + E^(8*x)),x]

[Out]

-x + Log[1 - 5*E^x + E^(9*x)]

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2320

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {-1+8 x^9}{x \left (1-5 x+x^9\right )} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {-5+9 x^8}{1-5 x+x^9}\right ) \, dx,x,e^x\right ) \\ & = -x+\text {Subst}\left (\int \frac {-5+9 x^8}{1-5 x+x^9} \, dx,x,e^x\right ) \\ & = -x+\log \left (1-5 e^x+e^{9 x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx=-\log \left (e^x\right )+\log \left (1-5 e^x+e^{9 x}\right ) \]

[In]

Integrate[(-E^(-x) + 8*E^(8*x))/(-5 + E^(-x) + E^(8*x)),x]

[Out]

-Log[E^x] + Log[1 - 5*E^x + E^(9*x)]

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\ln \left ({\mathrm e}^{8 x}+{\mathrm e}^{-x}-5\right )\)	\(12\)
default	\(\ln \left ({\mathrm e}^{8 x}+{\mathrm e}^{-x}-5\right )\)	\(12\)
parallelrisch	\(\ln \left ({\mathrm e}^{8 x}+{\mathrm e}^{-x}-5\right )\)	\(12\)
risch	\(8 x +\ln \left ({\mathrm e}^{-9 x}-5 \,{\mathrm e}^{-8 x}+1\right )\)	\(18\)
norman	\(8 x +\ln \left ({\mathrm e}^{-9 x}-5 \,{\mathrm e}^{-8 x}+1\right )\)	\(22\)

[In]

int((8*exp(8*x)-exp(-x))/(exp(8*x)+exp(-x)-5),x,method=_RETURNVERBOSE)

[Out]

ln(exp(8*x)+exp(-x)-5)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx=8 \, x + \log \left (-5 \, e^{\left (-8 \, x\right )} + e^{\left (-9 \, x\right )} + 1\right ) \]

[In]

integrate((8*exp(8*x)-exp(-x))/(exp(8*x)+exp(-x)-5),x, algorithm="fricas")

[Out]

8*x + log(-5*e^(-8*x) + e^(-9*x) + 1)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx=- x + \log {\left (e^{9 x} - 5 e^{x} + 1 \right )} \]

[In]

integrate((8*exp(8*x)-exp(-x))/(exp(8*x)+exp(-x)-5),x)

[Out]

-x + log(exp(9*x) - 5*exp(x) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx=\log \left (e^{\left (8 \, x\right )} + e^{\left (-x\right )} - 5\right ) \]

[In]

integrate((8*exp(8*x)-exp(-x))/(exp(8*x)+exp(-x)-5),x, algorithm="maxima")

[Out]

log(e^(8*x) + e^(-x) - 5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx=-x + \log \left ({\left | e^{\left (9 \, x\right )} - 5 \, e^{x} + 1 \right |}\right ) \]

[In]

integrate((8*exp(8*x)-exp(-x))/(exp(8*x)+exp(-x)-5),x, algorithm="giac")

[Out]

-x + log(abs(e^(9*x) - 5*e^x + 1))

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {-e^{-x}+8 e^{8 x}}{-5+e^{-x}+e^{8 x}} \, dx=\ln \left ({\mathrm {e}}^{9\,x}-5\,{\mathrm {e}}^x+1\right )-x \]

[In]

int(-(exp(-x) - 8*exp(8*x))/(exp(-x) + exp(8*x) - 5),x)

[Out]

log(exp(9*x) - 5*exp(x) + 1) - x