3.74 \(\int \frac{1}{\sqrt{1+\coth (x)}} \, dx\)

Optimal. Leaf size=32 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{1}{\sqrt{\coth (x)+1}} \]

[Out]

ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Coth[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0228691, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3479, 3480, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}}-\frac{1}{\sqrt{\coth (x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + Coth[x]],x]

[Out]

ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Coth[x]]

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 3480

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

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

Mathematica [C]  time = 0.291743, size = 51, normalized size = 1.59 \[ \frac{-2+(-1-i) \sqrt{i (\coth (x)+1)} \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i (\coth (x)+1)}\right )}{2 \sqrt{\coth (x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Coth[x]],x]

[Out]

(-2 - (1 + I)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]]*Sqrt[I*(1 + Coth[x])])/(2*Sqrt[1 + Coth[x]])

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 27, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1+{\rm coth} \left (x\right )}} \right ) }-{\frac{1}{\sqrt{1+{\rm coth} \left (x\right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+coth(x))^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\coth \left (x\right ) + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(coth(x) + 1), x)

________________________________________________________________________________________

Fricas [B]  time = 2.36078, size = 300, normalized size = 9.38 \begin{align*} \frac{{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \log \left (2 \, \sqrt{2} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right ) - 4 \, \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}}{4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^(1/2),x, algorithm="fricas")

[Out]

1/4*((sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*log(2*sqrt(2)*sqrt(sinh(x)/(cosh(x) - sinh(x)))*(cosh(x) + sinh(x)) +
2*cosh(x)^2 + 4*cosh(x)*sinh(x) + 2*sinh(x)^2 - 1) - 4*sqrt(sinh(x)/(cosh(x) - sinh(x))))/(cosh(x) + sinh(x))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))**(1/2),x)

[Out]

Integral(1/sqrt(coth(x) + 1), x)

________________________________________________________________________________________

Giac [B]  time = 1.15375, size = 89, normalized size = 2.78 \begin{align*} \frac{\sqrt{2}{\left (\frac{2}{\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}} - \log \left ({\left | 2 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )}}{4 \, \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(2)*(2/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x)) - log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1)))/sgn(
e^(2*x) - 1)