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]

1/17*arctanh(1/17*(4-exp(x))*17^(1/2))*17^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 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])

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 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]]]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.40, size = 42, normalized size = 2.10 \[ \frac {1}{34} \, \sqrt {17} \log \left (-\frac {2 \, {\left (\sqrt {17} + 4\right )} e^{x} - 8 \, \sqrt {17} - e^{\left (2 \, x\right )} - 33}{e^{\left (2 \, x\right )} - 8 \, e^{x} - 1}\right ) \]

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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giac [B]  time = 0.23, size = 33, normalized size = 1.65 \[ \frac {1}{34} \, \sqrt {17} \log \left (\frac {{\left | -2 \, \sqrt {17} + 2 \, e^{x} - 8 \right |}}{{\left | 2 \, \sqrt {17} + 2 \, e^{x} - 8 \right |}}\right ) \]

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))

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maple [A]  time = 0.03, size = 18, normalized size = 0.90 \[ -\frac {\sqrt {17}\, \arctanh \left (\frac {\left (2 \,{\mathrm e}^{x}-8\right ) \sqrt {17}}{34}\right )}{17} \]

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 = 2.07, size = 26, normalized size = 1.30 \[ \frac {1}{34} \, \sqrt {17} \log \left (-\frac {\sqrt {17} - e^{x} + 4}{\sqrt {17} + e^{x} - 4}\right ) \]

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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mupad [B]  time = 0.39, size = 17, normalized size = 0.85 \[ -\frac {\sqrt {17}\,\mathrm {atanh}\left (\frac {\sqrt {17}\,\left (2\,{\mathrm {e}}^x-8\right )}{34}\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.13, size = 17, normalized size = 0.85 \[ \operatorname {RootSum} {\left (68 z^{2} - 1, \left (i \mapsto i \log {\left (- 34 i + e^{x} - 4 \right )} \right )\right )} \]

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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