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

Optimal. Leaf size=17 \[ \log \left (5-e^{-x}-e^{8 x}\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2282, 6742, 1587} \begin {gather*} \log \left (-5 e^x+e^{9 x}+1\right )-x \end {gather*}

Antiderivative was successfully verified.

[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 1587

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]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, 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} &=\operatorname {Subst}\left (\int \frac {-1+8 x^9}{x \left (1-5 x+x^9\right )} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{x}+\frac {-5+9 x^8}{1-5 x+x^9}\right ) \, dx,x,e^x\right )\\ &=-x+\operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 17, normalized size = 1.00 \begin {gather*} -x+\log \left (1-5 e^x+e^{9 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x + Log[1 - 5*E^x + E^(9*x)]

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fricas [A]  time = 0.60, size = 17, normalized size = 1.00 \begin {gather*} 8 \, x + \log \left (-5 \, e^{\left (-8 \, x\right )} + e^{\left (-9 \, x\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.24, size = 16, normalized size = 0.94 \begin {gather*} -x + \log \left ({\left | e^{\left (9 \, x\right )} - 5 \, e^{x} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.04, size = 12, normalized size = 0.71

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.65 \begin {gather*} \log \left (e^{\left (8 \, x\right )} + e^{\left (-x\right )} - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.09, size = 15, normalized size = 0.88 \begin {gather*} \ln \left ({\mathrm {e}}^{9\,x}-5\,{\mathrm {e}}^x+1\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.13, size = 19, normalized size = 1.12 \begin {gather*} 8 x + \log {\left (1 - 5 e^{- 8 x} + e^{- 9 x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*x + log(1 - 5*exp(-8*x) + exp(-9*x))

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