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

Optimal. Leaf size=30 \[ \frac{1}{\sqrt{\coth (x)+1}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0386308, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3526, 3480, 206} \[ \frac{1}{\sqrt{\coth (x)+1}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/Sqrt[1 + Coth[x]],x]

[Out]

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

Rule 3526

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^m)/(2*a*f*m), x] + Dist[(b*c + a*d)/(2*a*b), Int[(a + b*Tan[e + f*x])^(m + 1),
x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 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{\coth (x)}{\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 = 0.194822, size = 97, normalized size = 3.23 \[ \frac{\text{csch}(x) \left (\frac{1}{2} \sinh (2 x)-\frac{1}{2} \cosh (2 x)+\frac{1}{2}\right ) (\sinh (x)+\cosh (x))}{\sqrt{\coth (x)+1}}+\frac{\left (\frac{1}{2}-\frac{i}{2}\right ) \text{csch}(x) (\sinh (x)+\cosh (x)) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i \coth (x)+i}\right )}{\sqrt{i \coth (x)+i} \sqrt{\coth (x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/Sqrt[1 + Coth[x]],x]

[Out]

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

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Maple [A]  time = 0.039, size = 25, 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(coth(x)/(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth(x)/sqrt(coth(x) + 1), x)

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Fricas [B]  time = 2.99012, size = 300, normalized size = 10. \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(coth(x)/(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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(x)/sqrt(coth(x) + 1), x)

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Giac [B]  time = 1.16177, size = 119, normalized size = 3.97 \begin{align*} -\frac{1}{2} \, \sqrt{2} \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - \frac{\sqrt{2} \log \left ({\left | -2 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} - 1 \right |}\right )}{4 \, \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac{\sqrt{2}}{2 \,{\left (\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )} \mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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