3.132 \(\int \coth (x) (1+\coth (x))^{3/2} \, dx\)

Optimal. Leaf size=45 \[ -\frac{2}{3} (\coth (x)+1)^{3/2}-2 \sqrt{\coth (x)+1}+2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.051792, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3527, 3478, 3480, 206} \[ -\frac{2}{3} (\coth (x)+1)^{3/2}-2 \sqrt{\coth (x)+1}+2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]*(1 + Coth[x])^(3/2),x]

[Out]

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

Rule 3527

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

Rule 3478

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Dist[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] && G
tQ[n, 1]

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 \coth (x) (1+\coth (x))^{3/2} \, dx &=-\frac{2}{3} (1+\coth (x))^{3/2}+\int (1+\coth (x))^{3/2} \, dx\\ &=-2 \sqrt{1+\coth (x)}-\frac{2}{3} (1+\coth (x))^{3/2}+2 \int \sqrt{1+\coth (x)} \, dx\\ &=-2 \sqrt{1+\coth (x)}-\frac{2}{3} (1+\coth (x))^{3/2}+4 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\coth (x)}\right )\\ &=2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1+\coth (x)}}{\sqrt{2}}\right )-2 \sqrt{1+\coth (x)}-\frac{2}{3} (1+\coth (x))^{3/2}\\ \end{align*}

Mathematica [C]  time = 0.135816, size = 90, normalized size = 2. \[ -\frac{2 (\coth (x)+1)^{3/2} \left (\cosh (x) \sqrt{i (\coth (x)+1)}+4 \sinh (x) \sqrt{i (\coth (x)+1)}-(3-3 i) \sinh (x) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i (\coth (x)+1)}\right )\right )}{3 \sqrt{i (\coth (x)+1)} (\sinh (x)+\cosh (x))} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]*(1 + Coth[x])^(3/2),x]

[Out]

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

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Maple [A]  time = 0.014, size = 35, normalized size = 0.8 \begin{align*} -{\frac{2}{3} \left ( 1+{\rm coth} \left (x\right ) \right ) ^{{\frac{3}{2}}}}+2\,{\it Artanh} \left ( 1/2\,\sqrt{1+{\rm coth} \left (x\right )}\sqrt{2} \right ) \sqrt{2}-2\,\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))^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\coth \left (x\right ) + 1\right )}^{\frac{3}{2}} \coth \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)*(1+coth(x))^(3/2),x, algorithm="maxima")

[Out]

integrate((coth(x) + 1)^(3/2)*coth(x), x)

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Fricas [B]  time = 3.052, size = 869, normalized size = 19.31 \begin{align*} -\frac{2 \, \sqrt{2}{\left (5 \, \sqrt{2} \cosh \left (x\right )^{3} + 15 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 5 \, \sqrt{2} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \sqrt{2} \cosh \left (x\right )^{2} - \sqrt{2}\right )} \sinh \left (x\right ) - 3 \, \sqrt{2} \cosh \left (x\right )\right )} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \,{\left (\sqrt{2} \cosh \left (x\right )^{4} + 4 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt{2} \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \sqrt{2} \cosh \left (x\right )^{2} - \sqrt{2}\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt{2} \cosh \left (x\right )^{2} + 4 \,{\left (\sqrt{2} \cosh \left (x\right )^{3} - \sqrt{2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt{2}\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 )}{3 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)*(1+coth(x))^(3/2),x, algorithm="fricas")

[Out]

-1/3*(2*sqrt(2)*(5*sqrt(2)*cosh(x)^3 + 15*sqrt(2)*cosh(x)*sinh(x)^2 + 5*sqrt(2)*sinh(x)^3 + 3*(5*sqrt(2)*cosh(
x)^2 - sqrt(2))*sinh(x) - 3*sqrt(2)*cosh(x))*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*cosh(x)*sinh(x) + 2*sinh(x)^2 - 1))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + si
nh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\coth{\left (x \right )} + 1\right )^{\frac{3}{2}} \coth{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)*(1+coth(x))**(3/2),x)

[Out]

Integral((coth(x) + 1)**(3/2)*coth(x), x)

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Giac [B]  time = 1.18529, size = 182, normalized size = 4.04 \begin{align*} -\frac{1}{3} \, \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{2 \,{\left (9 \,{\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 ) + 12 \,{\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 ) + 5 \, \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)*(1+coth(x))^(3/2),x, algorithm="giac")

[Out]

-1/3*sqrt(2)*(3*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))*sgn(e^(2*x) - 1) + 2*(9*(sqrt(e^(4*x) - e^
(2*x)) - e^(2*x))^2*sgn(e^(2*x) - 1) + 12*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))*sgn(e^(2*x) - 1) + 5*sgn(e^(2*x)
- 1))/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x) + 1)^3)