3.524 \(\int \frac{-1+e^x}{1+e^x} \, dx\)

Optimal. Leaf size=12 \[ 2 \log \left (e^x+1\right )-x \]

[Out]

-x + 2*Log[1 + E^x]

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Rubi [A]  time = 0.0218481, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2282, 72} \[ 2 \log \left (e^x+1\right )-x \]

Antiderivative was successfully verified.

[In]

Int[(-1 + E^x)/(1 + E^x),x]

[Out]

-x + 2*Log[1 + E^x]

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

Mathematica [A]  time = 0.0081372, size = 12, normalized size = 1. \[ 2 \log \left (e^x+1\right )-x \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^x)/(1 + E^x),x]

[Out]

-x + 2*Log[1 + E^x]

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Maple [A]  time = 0.005, size = 14, normalized size = 1.2 \begin{align*} -\ln \left ({{\rm e}^{x}} \right ) +2\,\ln \left ( 1+{{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.927006, size = 15, normalized size = 1.25 \begin{align*} -x + 2 \, \log \left (e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log(e^x + 1)

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Fricas [A]  time = 1.86974, size = 28, normalized size = 2.33 \begin{align*} -x + 2 \, \log \left (e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log(e^x + 1)

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Sympy [A]  time = 0.077769, size = 8, normalized size = 0.67 \begin{align*} - x + 2 \log{\left (e^{x} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log(exp(x) + 1)

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Giac [A]  time = 1.11951, size = 15, normalized size = 1.25 \begin{align*} -x + 2 \, \log \left (e^{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log(e^x + 1)