3.138 \(\int \frac{\coth ^2(x)}{\sqrt{1+\coth (x)}} \, dx\)

Optimal. Leaf size=42 \[ -2 \sqrt{\coth (x)+1}-\frac{1}{\sqrt{\coth (x)+1}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.058539, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3543, 3479, 3480, 206} \[ -2 \sqrt{\coth (x)+1}-\frac{1}{\sqrt{\coth (x)+1}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2/Sqrt[1 + Coth[x]],x]

[Out]

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

Rule 3543

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
(d^2*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e
+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1] &&  !(EqQ[m, 2] && EqQ
[a, 0])

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

Mathematica [C]  time = 0.359309, size = 81, normalized size = 1.93 \[ \frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \text{csch}(x) (\sinh (x)+\cosh (x)) \left (\left (\frac{1}{2}-\frac{i}{2}\right ) (-\sinh (2 x)+\cosh (2 x)-5)-\frac{i \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i (\coth (x)+1)}\right )}{\sqrt{i (\coth (x)+1)}}\right )}{\sqrt{\coth (x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^2/Sqrt[1 + Coth[x]],x]

[Out]

((1/2 + I/2)*Csch[x]*(Cosh[x] + Sinh[x])*(((-I)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]])/Sqrt[I*(1 + Coth[x]
)] + (1/2 - I/2)*(-5 + Cosh[2*x] - Sinh[2*x])))/Sqrt[1 + Coth[x]]

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Maple [A]  time = 0.041, size = 35, 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 )}}}-2\,\sqrt{1+{\rm coth} \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2/(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)-2*(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 )^{2}}{\sqrt{\coth \left (x\right ) + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(coth(x)^2/sqrt(coth(x) + 1), x)

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Fricas [B]  time = 3.11865, size = 637, normalized size = 15.17 \begin{align*} -\frac{2 \, \sqrt{2}{\left (5 \, \sqrt{2} \cosh \left (x\right )^{2} + 10 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + 5 \, \sqrt{2} \sinh \left (x\right )^{2} - \sqrt{2}\right )} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} -{\left (\sqrt{2} \cosh \left (x\right )^{3} + 3 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt{2} \sinh \left (x\right )^{3} +{\left (3 \, \sqrt{2} \cosh \left (x\right )^{2} - \sqrt{2}\right )} \sinh \left (x\right ) - \sqrt{2} \cosh \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 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*(5*sqrt(2)*cosh(x)^2 + 10*sqrt(2)*cosh(x)*sinh(x) + 5*sqrt(2)*sinh(x)^2 - sqrt(2))*sqrt(sinh(x
)/(cosh(x) - sinh(x))) - (sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sqrt(2)*sinh(x)^3 + (3*sqrt(2)*cos
h(x)^2 - sqrt(2))*sinh(x) - sqrt(2)*cosh(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))/(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cos
h(x)^2 - 1)*sinh(x) - cosh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\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)**2/(1+coth(x))**(1/2),x)

[Out]

Integral(coth(x)**2/sqrt(coth(x) + 1), x)

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Giac [B]  time = 1.20583, size = 119, normalized size = 2.83 \begin{align*} -\frac{\frac{5 \, \sqrt{2} e^{\left (2 \, x\right )}}{\mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac{\sqrt{2}}{\mathrm{sgn}\left (e^{\left (2 \, x\right )} - 1\right )}}{2 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(5*sqrt(2)*e^(2*x)/sgn(e^(2*x) - 1) - sqrt(2)/sgn(e^(2*x) - 1))/sqrt(e^(4*x) - e^(2*x)) - 1/4*sqrt(2)*log
(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))/sgn(e^(2*x) - 1)