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

Optimal. Leaf size=34 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-\frac {2}{3} (\coth (x)+1)^{3/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3543, 3480, 206} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-\frac {2}{3} (\coth (x)+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2*Sqrt[1 + Coth[x]],x]

[Out]

Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - (2*(1 + Coth[x])^(3/2))/3

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 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 \coth ^2(x) \sqrt {1+\coth (x)} \, dx &=-\frac {2}{3} (1+\coth (x))^{3/2}+\int \sqrt {1+\coth (x)} \, dx\\ &=-\frac {2}{3} (1+\coth (x))^{3/2}+2 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-\frac {2}{3} (1+\coth (x))^{3/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.18, size = 61, normalized size = 1.79 \[ \frac {-2 \coth ^2(x)-4 \coth (x)-(3+3 i) \sqrt {i (\coth (x)+1)} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )-2}{3 \sqrt {\coth (x)+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2 - 4*Coth[x] - 2*Coth[x]^2 - (3 + 3*I)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]]*Sqrt[I*(1 + Coth[x])])/(3*
Sqrt[1 + Coth[x]])

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 242, normalized size = 7.12 \[ -\frac {8 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x)^{3} + 3 \, \sqrt {2} \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{2} + \sqrt {2} \sinh \relax (x)^{3}\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 3 \, {\left (\sqrt {2} \cosh \relax (x)^{4} + 4 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{3} + \sqrt {2} \sinh \relax (x)^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {2} \cosh \relax (x)^{2} + 4 \, {\left (\sqrt {2} \cosh \relax (x)^{3} - \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) + \sqrt {2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )}{6 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(8*sqrt(2)*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)^2*sinh(x) + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sqrt(2)*sinh(
x)^3)*sqrt(sinh(x)/(cosh(x) - sinh(x))) - 3*(sqrt(2)*cosh(x)^4 + 4*sqrt(2)*cosh(x)*sinh(x)^3 + sqrt(2)*sinh(x)
^4 + 2*(3*sqrt(2)*cosh(x)^2 - sqrt(2))*sinh(x)^2 - 2*sqrt(2)*cosh(x)^2 + 4*(sqrt(2)*cosh(x)^3 - sqrt(2)*cosh(x
))*sinh(x) + sqrt(2))*log(2*sqrt(2)*sqrt(sinh(x)/(cosh(x) - sinh(x)))*(cosh(x) + sinh(x)) + 2*cosh(x)^2 + 4*co
sh(x)*sinh(x) + 2*sinh(x)^2 - 1))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2
- 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)

________________________________________________________________________________________

giac [B]  time = 0.16, size = 133, normalized size = 3.91 \[ -\frac {1}{6} \, \sqrt {2} {\left (3 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {8 \, {\left (3 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 3 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(2)*(3*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))*sgn(e^(2*x) - 1) + 8*(3*(sqrt(e^(4*x) - e^
(2*x)) - e^(2*x))^2*sgn(e^(2*x) - 1) + 3*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))*sgn(e^(2*x) - 1) + sgn(e^(2*x) -
1))/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x) + 1)^3)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 26, normalized size = 0.76 \[ -\frac {2 \left (1+\coth \relax (x )\right )^{\frac {3}{2}}}{3}+\arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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(sqrt(coth(x) + 1)*coth(x)^2, x)

________________________________________________________________________________________

mupad [B]  time = 1.19, size = 25, normalized size = 0.74 \[ \sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )-\frac {2\,{\left (\mathrm {coth}\relax (x)+1\right )}^{3/2}}{3} \]

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*(coth(x) + 1)^(3/2))/3

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________