3.227 \(\int e^{-x} \cot ^{-1}(e^x) \, dx\)

Optimal. Leaf size=27 \[ -x+\frac {1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2194, 5208, 2282, 36, 29, 31} \[ -x+\frac {1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[E^x]/E^x,x]

[Out]

-x - ArcCot[E^x]/E^x + Log[1 + E^(2*x)]/2

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, 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 5208

Int[((a_.) + ArcCot[u_]*(b_.))*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[a + b*ArcCot[u], w, x] + Dist
[b, Int[SimplifyIntegrand[(w*D[u, x])/(1 + u^2), x], x], x] /; InverseFunctionFreeQ[w, x]] /; FreeQ[{a, b}, x]
&& InverseFunctionFreeQ[u, x] &&  !MatchQ[v, ((c_.) + (d_.)*x)^(m_.) /; FreeQ[{c, d, m}, x]] && FalseQ[Functi
onOfLinear[v*(a + b*ArcCot[u]), x]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.00 \[ -x+\frac {1}{2} \log \left (e^{2 x}+1\right )-e^{-x} \cot ^{-1}\left (e^x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[E^x]/E^x,x]

[Out]

-x - ArcCot[E^x]/E^x + Log[1 + E^(2*x)]/2

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fricas [A]  time = 0.52, size = 28, normalized size = 1.04 \[ -\frac {1}{2} \, {\left (2 \, x e^{x} - e^{x} \log \left (e^{\left (2 \, x\right )} + 1\right ) + 2 \, \operatorname {arccot}\left (e^{x}\right )\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(exp(x))/exp(x),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.12, size = 21, normalized size = 0.78 \[ -\arctan \left (e^{\left (-x\right )}\right ) e^{\left (-x\right )} + \frac {1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(exp(x))/exp(x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 25, normalized size = 0.93 \[ -\mathrm {arccot}\left ({\mathrm e}^{x}\right ) {\mathrm e}^{-x}-\ln \left ({\mathrm e}^{x}\right )+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(exp(x))/exp(x),x)

[Out]

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

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maxima [A]  time = 0.34, size = 19, normalized size = 0.70 \[ -\operatorname {arccot}\left (e^{x}\right ) e^{\left (-x\right )} + \frac {1}{2} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(exp(x))/exp(x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.09, size = 22, normalized size = 0.81 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )}{2}-x-\mathrm {acot}\left ({\mathrm {e}}^x\right )\,{\mathrm {e}}^{-x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(exp(x))*exp(-x),x)

[Out]

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

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sympy [A]  time = 6.09, size = 19, normalized size = 0.70 \[ - x + \frac {\log {\left (e^{2 x} + 1 \right )}}{2} - e^{- x} \operatorname {acot}{\left (e^{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(exp(x))/exp(x),x)

[Out]

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

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