3.8.18 \(\int \frac {1}{e^{-x}+e^x} \, dx\) [718]

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

[Out]

arctan(exp(x))

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Rubi [A]
time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2320, 209} \begin {gather*} \text {ArcTan}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-x) + E^x)^(-1),x]

[Out]

ArcTan[E^x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 {1}{e^{-x}+e^x} \, dx &=\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=\tan ^{-1}\left (e^x\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 4, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-x) + E^x)^(-1),x]

[Out]

ArcTan[E^x]

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Maple [A]
time = 0.02, size = 4, normalized size = 1.00

method	result	size
default	\(\arctan \left ({\mathrm e}^{x}\right )\)	\(4\)
risch	\(\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(exp(x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 7 vs. \(2 (3) = 6\).
time = 0.52, size = 7, normalized size = 1.75 \begin {gather*} -\arctan \left (e^{\left (-x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(e^(-x))

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Fricas [A]
time = 0.36, size = 3, normalized size = 0.75 \begin {gather*} \arctan \left (e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(e^x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (3) = 6\).
time = 0.03, size = 15, normalized size = 3.75 \begin {gather*} \operatorname {RootSum} {\left (4 z^{2} + 1, \left ( i \mapsto i \log {\left (2 i + e^{x} \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + exp(x))))

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Giac [A]
time = 5.94, size = 3, normalized size = 0.75 \begin {gather*} \arctan \left (e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(e^x)

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Mupad [B]
time = 0.02, size = 3, normalized size = 0.75 \begin {gather*} \mathrm {atan}\left ({\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(exp(-x) + exp(x)),x)

[Out]

atan(exp(x))

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