3.8.28 \(\int \frac {1}{-e^x+e^{3 x}} \, dx\) [728]

Optimal. Leaf size=12 \[ e^{-x}-\tanh ^{-1}\left (e^x\right ) \]

[Out]

exp(-x)-arctanh(exp(x))

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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2320, 331, 213} \begin {gather*} e^{-x}-\tanh ^{-1}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-E^x + E^(3*x))^(-1),x]

[Out]

E^(-x) - ArcTanh[E^x]

Rule 213

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, 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]]]

Rubi steps

\begin {align*} \int \frac {1}{-e^x+e^{3 x}} \, dx &=\text {Subst}\left (\int \frac {1}{x^2 \left (-1+x^2\right )} \, dx,x,e^x\right )\\ &=e^{-x}+\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^x\right )\\ &=e^{-x}-\tanh ^{-1}\left (e^x\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 12, normalized size = 1.00 \begin {gather*} e^{-x}-\tanh ^{-1}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-E^x + E^(3*x))^(-1),x]

[Out]

E^(-x) - ArcTanh[E^x]

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Maple [A]
time = 0.02, size = 20, normalized size = 1.67

method	result	size
default	\(-\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}+\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}+{\mathrm e}^{-x}\)	\(20\)
norman	\(-\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}+\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}+{\mathrm e}^{-x}\)	\(20\)
risch	\(-\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}+\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}+{\mathrm e}^{-x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-exp(x)+exp(3*x)),x,method=_RETURNVERBOSE)

[Out]

-1/2*ln(1+exp(x))+1/2*ln(-1+exp(x))+1/exp(x)

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Maxima [A]
time = 0.30, size = 19, normalized size = 1.58 \begin {gather*} e^{\left (-x\right )} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left (e^{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-exp(x)+exp(3*x)),x, algorithm="maxima")

[Out]

e^(-x) - 1/2*log(e^x + 1) + 1/2*log(e^x - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (10) = 20\).
time = 0.39, size = 25, normalized size = 2.08 \begin {gather*} -\frac {1}{2} \, {\left (e^{x} \log \left (e^{x} + 1\right ) - e^{x} \log \left (e^{x} - 1\right ) - 2\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-exp(x)+exp(3*x)),x, algorithm="fricas")

[Out]

-1/2*(e^x*log(e^x + 1) - e^x*log(e^x - 1) - 2)*e^(-x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).
time = 0.04, size = 20, normalized size = 1.67 \begin {gather*} \frac {\log {\left (e^{x} - 1 \right )}}{2} - \frac {\log {\left (e^{x} + 1 \right )}}{2} + e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-exp(x)+exp(3*x)),x)

[Out]

log(exp(x) - 1)/2 - log(exp(x) + 1)/2 + exp(-x)

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Giac [A]
time = 6.28, size = 20, normalized size = 1.67 \begin {gather*} e^{\left (-x\right )} - \frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-exp(x)+exp(3*x)),x, algorithm="giac")

[Out]

e^(-x) - 1/2*log(e^x + 1) + 1/2*log(abs(e^x - 1))

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Mupad [B]
time = 0.06, size = 19, normalized size = 1.58 \begin {gather*} {\mathrm {e}}^{-x}+\frac {\ln \left ({\mathrm {e}}^x-1\right )}{2}-\frac {\ln \left ({\mathrm {e}}^x+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(exp(3*x) - exp(x)),x)

[Out]

exp(-x) + log(exp(x) - 1)/2 - log(exp(x) + 1)/2

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