3.20 \(\int \sqrt{1+e^{-x}} \text{csch}(x) \, dx\)

Optimal. Leaf size=25 \[ -2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{e^{-x}+1}}{\sqrt{2}}\right ) \]

[Out]

-2*Sqrt[2]*ArcTanh[Sqrt[1 + E^(-x)]/Sqrt[2]]

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Rubi [A]  time = 0.0510541, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2282, 12, 1446, 1469, 627, 63, 206} \[ -2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{e^{-x}+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + E^(-x)]*Csch[x],x]

[Out]

-2*Sqrt[2]*ArcTanh[Sqrt[1 + E^(-x)]/Sqrt[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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1446

Int[((a_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Int[((d + e*x^n)^q*(c + a
*x^(2*n))^p)/x^(2*n*p), x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]

Rule 1469

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Sim
plify[m - n + 1], 0]

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

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

Rubi steps

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

Mathematica [B]  time = 0.0949289, size = 126, normalized size = 5.04 \[ \frac{\sqrt{2} e^{x/2} \sqrt{e^{-x}+1} \left (\log \left (1-e^{-x/2}\right )+\log \left (e^{-x/2}+1\right )-\log \left (e^{-x/2} \left (\sqrt{2} \sqrt{e^x+1}+e^{x/2}-1\right )\right )-\log \left (e^{-x/2} \left (\sqrt{2} \sqrt{e^x+1}+e^{x/2}+1\right )\right )\right )}{\sqrt{e^x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + E^(-x)]*Csch[x],x]

[Out]

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

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Maple [A]  time = 0.071, size = 33, normalized size = 1.3 \begin{align*} -2\,\sqrt{2}\sqrt{ \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-1}}\sqrt{\tanh \left ( x/2 \right ) +1}{\it Artanh} \left ( \sqrt{\tanh \left ( x/2 \right ) +1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+exp(-x))^(1/2)/sinh(x),x)

[Out]

-2*2^(1/2)*(1/(tanh(1/2*x)+1))^(1/2)*(tanh(1/2*x)+1)^(1/2)*arctanh((tanh(1/2*x)+1)^(1/2))

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Maxima [A]  time = 1.4304, size = 47, normalized size = 1.88 \begin{align*} \sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{e^{\left (-x\right )} + 1}}{\sqrt{2} + \sqrt{e^{\left (-x\right )} + 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.23759, size = 211, normalized size = 8.44 \begin{align*} \sqrt{2} \log \left (\frac{2 \,{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\frac{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right )}} - 3 \, \cosh \left (x\right ) - 3 \, \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*log((2*(sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt((cosh(x) + sinh(x) + 1)/(cosh(x) + sinh(x))) - 3*cosh(
x) - 3*sinh(x) - 1)/(cosh(x) + sinh(x) - 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{1 + e^{- x}}}{\sinh{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 + exp(-x))/sinh(x), x)

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Giac [B]  time = 1.26869, size = 100, normalized size = 4. \begin{align*} -\sqrt{2} \log \left (\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right ) + \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}{{\left | 2 \, \sqrt{2} + 2 \, \sqrt{e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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