3.8.6 \(\int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx\) [706]

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 = 0.03, 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 = {2320, 632, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {4-e^x}{\sqrt {17}}\right )}{\sqrt {17}} \end {gather*}

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 212

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

Rule 632

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 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]]]

Rubi steps

\begin {align*} \int \frac {e^x}{-1-8 e^x+e^{2 x}} \, dx &=\text {Subst}\left (\int \frac {1}{-1-8 x+x^2} \, dx,x,e^x\right )\\ &=-\left (2 \text {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 = 0.05, size = 19, normalized size = 0.95 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {-4+e^x}{\sqrt {17}}\right )}{\sqrt {17}} \end {gather*}

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 = 0.02, size = 18, normalized size = 0.90

method	result	size
default	\(-\frac {\sqrt {17}\, \arctanh \left (\frac {\left (2 \,{\mathrm e}^{x}-8\right ) \sqrt {17}}{34}\right )}{17}\)	\(18\)
risch	\(\frac {\sqrt {17}\, \ln \left ({\mathrm e}^{x}-4-\sqrt {17}\right )}{34}-\frac {\sqrt {17}\, \ln \left ({\mathrm e}^{x}-4+\sqrt {17}\right )}{34}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 26, normalized size = 1.30 \begin {gather*} \frac {1}{34} \, \sqrt {17} \log \left (-\frac {\sqrt {17} - e^{x} + 4}{\sqrt {17} + e^{x} - 4}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
time = 0.34, size = 42, normalized size = 2.10 \begin {gather*} \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 {gather*}

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 = 0.04, size = 17, normalized size = 0.85 \begin {gather*} \operatorname {RootSum} {\left (68 z^{2} - 1, \left ( i \mapsto i \log {\left (- 34 i + e^{x} - 4 \right )} \right )\right )} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
time = 4.59, size = 33, normalized size = 1.65 \begin {gather*} \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 {gather*}

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

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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