∫ √ ____ 1 + e−x csch(x) dx [20]

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

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2320, 12, 1460, 1483, 641, 65, 212} \begin {gather*} -2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {e^{-x}+1}}{\sqrt {2}}\right ) \end {gather*}

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

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

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,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^

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

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]

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

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

\begin {align*} \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx &=\text {Subst}\left (\int \frac {2 \sqrt {1+\frac {1}{x}}}{-1+x^2} \, dx,x,e^x\right )\\ &=2 \text {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}}}{-1+x^2} \, dx,x,e^x\right )\\ &=2 \text {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 \text {Subst}\left (\int \frac {\sqrt {1+x}}{1-x^2} \, dx,x,e^{-x}\right )\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x}} \, dx,x,e^{-x}\right )\right )\\ &=-\left (4 \text {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] Leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(25)=50\).
time = 0.08, size = 126, normalized size = 5.04 \begin {gather*} \frac {\sqrt {2} e^{x/2} \sqrt {1+e^{-x}} \left (\log \left (1-e^{-x/2}\right )+\log \left (1+e^{-x/2}\right )-\log \left (e^{-x/2} \left (-1+e^{x/2}+\sqrt {2} \sqrt {1+e^x}\right )\right )-\log \left (e^{-x/2} \left (1+e^{x/2}+\sqrt {2} \sqrt {1+e^x}\right )\right )\right )}{\sqrt {1+e^x}} \end {gather*}

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

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

________________________________________________________________________________________

Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

mathics('Integrate[Sqrt[1 + Exp[-x]]/Sinh[x],x]')

cought exception: maximum recursion depth exceeded while calling a Python object

________________________________________________________________________________________


method	result	size

default	\(-2 \sqrt {2}\, \sqrt {\frac {1}{\tanh \left (\frac {x}{2}\right )+1}}\, \sqrt {\tanh \left (\frac {x}{2}\right )+1}\, \arctanh \left (\sqrt {\tanh \left (\frac {x}{2}\right )+1}\right )\)	\(33\)

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

-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))

________________________________________________________________________________________

Maxima [A]
time = 0.39, size = 35, normalized size = 1.40 \begin {gather*} \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {e^{\left (-x\right )} + 1}}{\sqrt {2} + \sqrt {e^{\left (-x\right )} + 1}}\right ) \end {gather*}

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

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (19) = 38\).
time = 0.36, size = 55, normalized size = 2.20 \begin {gather*} \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 {gather*}

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

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))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 + e^{- x}}}{\sinh {\left (x \right )}}\, dx \end {gather*}

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (19) = 38\).
time = 0.02, size = 93, normalized size = 3.72 \begin {gather*} 2 \left (-\frac {\ln \left (\frac {\sqrt {2}-1}{\sqrt {2}+1}\right )}{\sqrt {2}}+\frac {\ln \left (\frac {\left |2 \left (-\mathrm {e}^{x}+\sqrt {\left (\mathrm {e}^{x}\right )^{2}+\mathrm {e}^{x}}\right )-2 \sqrt {2}+2\right |}{\left |2 \left (-\mathrm {e}^{x}+\sqrt {\left (\mathrm {e}^{x}\right )^{2}+\mathrm {e}^{x}}\right )+2 \sqrt {2}+2\right |}\right )}{\sqrt {2}}\right ) \end {gather*}

-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))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {{\mathrm {e}}^{-x}+1}}{\mathrm {sinh}\left (x\right )} \,d x \end {gather*}

________________________________________________________________________________________

3.1.20 \(\int \sqrt {1+e^{-x}} \text {csch}(x) \, dx\) [20]