3.706 \(\int \frac{e^x}{-1-8 e^x+e^{2 x}} \, dx\)

Optimal. Leaf size=20 \[ \frac{\tanh ^{-1}\left (\frac{4-e^x}{\sqrt{17}}\right )}{\sqrt{17}} \]

[Out]

ArcTanh[(4 - E^x)/Sqrt[17]]/Sqrt[17]

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Rubi [A]  time = 0.0430682, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{4-e^x}{\sqrt{17}}\right )}{\sqrt{17}} \]

Antiderivative was successfully verified.

[In]

Int[E^x/(-1 - 8*E^x + E^(2*x)),x]

[Out]

ArcTanh[(4 - E^x)/Sqrt[17]]/Sqrt[17]

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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{e^x}{-1-8 e^x+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{-1-8 x+x^2} \, dx,x,e^x\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{68-x^2} \, dx,x,-8+2 e^x\right )\right )\\ &=\frac{\tanh ^{-1}\left (\frac{4-e^x}{\sqrt{17}}\right )}{\sqrt{17}}\\ \end{align*}

Mathematica [A]  time = 0.0121242, size = 19, normalized size = 0.95 \[ -\frac{\tanh ^{-1}\left (\frac{e^x-4}{\sqrt{17}}\right )}{\sqrt{17}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(-1 - 8*E^x + E^(2*x)),x]

[Out]

-(ArcTanh[(-4 + E^x)/Sqrt[17]]/Sqrt[17])

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Maple [A]  time = 0.021, size = 18, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{17}}{17}{\it Artanh} \left ({\frac{ \left ( 2\,{{\rm e}^{x}}-8 \right ) \sqrt{17}}{34}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(-1-8*exp(x)+exp(2*x)),x)

[Out]

-1/17*17^(1/2)*arctanh(1/34*(2*exp(x)-8)*17^(1/2))

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Maxima [A]  time = 1.45928, size = 35, normalized size = 1.75 \begin{align*} \frac{1}{34} \, \sqrt{17} \log \left (-\frac{\sqrt{17} - e^{x} + 4}{\sqrt{17} + e^{x} - 4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(-1-8*exp(x)+exp(2*x)),x, algorithm="maxima")

[Out]

1/34*sqrt(17)*log(-(sqrt(17) - e^x + 4)/(sqrt(17) + e^x - 4))

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Fricas [B]  time = 0.916552, size = 127, normalized size = 6.35 \begin{align*} \frac{1}{34} \, \sqrt{17} \log \left (-\frac{2 \,{\left (\sqrt{17} + 4\right )} e^{x} - 8 \, \sqrt{17} - e^{\left (2 \, x\right )} - 33}{e^{\left (2 \, x\right )} - 8 \, e^{x} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(-1-8*exp(x)+exp(2*x)),x, algorithm="fricas")

[Out]

1/34*sqrt(17)*log(-(2*(sqrt(17) + 4)*e^x - 8*sqrt(17) - e^(2*x) - 33)/(e^(2*x) - 8*e^x - 1))

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Sympy [A]  time = 0.13306, size = 17, normalized size = 0.85 \begin{align*} \operatorname{RootSum}{\left (68 z^{2} - 1, \left ( i \mapsto i \log{\left (- 34 i + e^{x} - 4 \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(-1-8*exp(x)+exp(2*x)),x)

[Out]

RootSum(68*_z**2 - 1, Lambda(_i, _i*log(-34*_i + exp(x) - 4)))

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Giac [B]  time = 1.27252, size = 45, normalized size = 2.25 \begin{align*} \frac{1}{34} \, \sqrt{17} \log \left (\frac{{\left | -2 \, \sqrt{17} + 2 \, e^{x} - 8 \right |}}{{\left | 2 \, \sqrt{17} + 2 \, e^{x} - 8 \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(-1-8*exp(x)+exp(2*x)),x, algorithm="giac")

[Out]

1/34*sqrt(17)*log(abs(-2*sqrt(17) + 2*e^x - 8)/abs(2*sqrt(17) + 2*e^x - 8))