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

Optimal result

Integrand size = 14, antiderivative size = 25 \[ \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx=-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+e^{-x}}}{\sqrt {2}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2320, 12, 1460, 1483, 641, 65, 212} \[ \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx=-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {e^{-x}+1}}{\sqrt {2}}\right ) \]

[In]

Int[Sqrt[1 + E^(-x)]*Csch[x],x]

[Out]

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

Rule 12

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

Rule 65

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 212

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])

Rule 641

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
ntegerQ[m + p]))

Rule 1460

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 1483

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 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*} \text {integral}& = \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} \text {arctanh}\left (\frac {\sqrt {1+e^{-x}}}{\sqrt {2}}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(25)=50\).

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28 \[ \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx=-\frac {2 \sqrt {2} e^{x/2} \sqrt {1+e^{-x}} \text {arctanh}\left (\frac {\sqrt {2} e^{x/2}}{\sqrt {1+e^x}}\right )}{\sqrt {1+e^x}} \]

[In]

Integrate[Sqrt[1 + E^(-x)]*Csch[x],x]

[Out]

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

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

[In]

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

[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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (19) = 38\).

Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx=\sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right )}} - 3 \, \cosh \left (x\right ) - 3 \, \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) \]

[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))

Sympy [F]

\[ \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx=\int \frac {\sqrt {1 + e^{- x}}}{\sinh {\left (x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx=\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {e^{\left (-x\right )} + 1}}{\sqrt {2} + \sqrt {e^{\left (-x\right )} + 1}}\right ) \]

[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)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (19) = 38\).

Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx=\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx=\int \frac {\sqrt {{\mathrm {e}}^{-x}+1}}{\mathrm {sinh}\left (x\right )} \,d x \]

[In]

int((exp(-x) + 1)^(1/2)/sinh(x),x)

[Out]

int((exp(-x) + 1)^(1/2)/sinh(x), x)