3.656 \(\int \frac{e^{-2 x}+e^{2 x}}{-e^{-2 x}+e^{2 x}} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{2} \log \left (1-e^{4 x}\right )-x \]

[Out]

-x + Log[1 - E^(4*x)]/2

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Rubi [A]  time = 0.0444535, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2282, 446, 72} \[ \frac{1}{2} \log \left (1-e^{4 x}\right )-x \]

Antiderivative was successfully verified.

[In]

Int[(E^(-2*x) + E^(2*x))/(-E^(-2*x) + E^(2*x)),x]

[Out]

-x + Log[1 - E^(4*x)]/2

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.008311, size = 18, normalized size = 1. \[ \frac{1}{2} \log \left (1-e^{4 x}\right )-x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x + Log[1 - E^(4*x)]/2

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Maple [A]  time = 0.03, size = 30, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( 1+ \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{2}}+{\frac{\ln \left ( -1+{{\rm e}^{x}} \right ) }{2}}-\ln \left ({{\rm e}^{x}} \right ) +{\frac{\ln \left ( 1+{{\rm e}^{x}} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.98155, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} - e^{\left (-2 \, x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(e^(2*x) - e^(-2*x))

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Fricas [A]  time = 0.852482, size = 36, normalized size = 2. \begin{align*} -x + \frac{1}{2} \, \log \left (e^{\left (4 \, x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 1/2*log(e^(4*x) - 1)

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Sympy [A]  time = 0.095284, size = 10, normalized size = 0.56 \begin{align*} - x + \frac{\log{\left (e^{4 x} - 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(exp(4*x) - 1)/2

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Giac [A]  time = 1.20685, size = 19, normalized size = 1.06 \begin{align*} -x + \frac{1}{2} \, \log \left ({\left | e^{\left (4 \, x\right )} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 1/2*log(abs(e^(4*x) - 1))