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]

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)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 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])

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 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])

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 = 0.37, 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]]

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 189, normalized size = 4.50 \[ -\frac {2 \, \sqrt {2} {\left (5 \, \sqrt {2} \cosh \relax (x)^{2} + 10 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + 5 \, \sqrt {2} \sinh \relax (x)^{2} - \sqrt {2}\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - {\left (\sqrt {2} \cosh \relax (x)^{3} + 3 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{2} + \sqrt {2} \sinh \relax (x)^{3} + {\left (3 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x) - \sqrt {2} \cosh \relax (x)\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )}{4 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x) - \cosh \relax (x)\right )}} \]

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))

________________________________________________________________________________________

giac [B]  time = 0.18, size = 88, normalized size = 2.10 \[ -\frac {\frac {5 \, \sqrt {2} e^{\left (2 \, x\right )}}{\mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac {\sqrt {2}}{\mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )}}{2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}}} - \frac {\sqrt {2} \log \left ({\left | 4 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 4 \, e^{\left (2 \, x\right )} + 2 \right |}\right )}{4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]

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

________________________________________________________________________________________

maple [A]  time = 0.09, size = 35, normalized size = 0.83 \[ \frac {\arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {1}{\sqrt {1+\coth \relax (x )}}-2 \sqrt {1+\coth \relax (x )} \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \relax (x)^{2}}{\sqrt {\coth \relax (x) + 1}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.26, size = 36, normalized size = 0.86 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )}{2}-\frac {3}{\sqrt {\mathrm {coth}\relax (x)+1}}-\frac {2\,\mathrm {coth}\relax (x)}{\sqrt {\mathrm {coth}\relax (x)+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2/(coth(x) + 1)^(1/2),x)

[Out]

(2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2))/2 - 3/(coth(x) + 1)^(1/2) - (2*coth(x))/(coth(x) + 1)^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}{\relax (x )}}{\sqrt {\coth {\relax (x )} + 1}}\, dx \]

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)

________________________________________________________________________________________