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

Optimal. Leaf size=33 \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )-2 \sqrt{\coth (x)+1} \]

[Out]

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

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Rubi [A]  time = 0.0219855, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3478, 3480, 206} \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\coth (x)+1}}{\sqrt{2}}\right )-2 \sqrt{\coth (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Coth[x])^(3/2),x]

[Out]

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

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 (1+\coth (x))^{3/2} \, dx &=-2 \sqrt{1+\coth (x)}+2 \int \sqrt{1+\coth (x)} \, dx\\ &=-2 \sqrt{1+\coth (x)}+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)}\\ \end{align*}

Mathematica [C]  time = 0.106497, size = 69, normalized size = 2.09 \[ -\frac{2 \sinh (x) (\coth (x)+1)^{3/2} \left (\sqrt{i (\coth (x)+1)}-(1-i) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{i (\coth (x)+1)}\right )\right )}{\sqrt{i (\coth (x)+1)} (\sinh (x)+\cosh (x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Coth[x])^(3/2),x]

[Out]

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

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Maple [A]  time = 0.013, size = 27, normalized size = 0.8 \begin{align*} 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((1+coth(x))^(3/2),x)

[Out]

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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((coth(x) + 1)^(3/2), x)

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Fricas [B]  time = 2.48679, size = 450, normalized size = 13.64 \begin{align*} -\frac{2 \, \sqrt{2}{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\frac{\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} -{\left (\sqrt{2} \cosh \left (x\right )^{2} + 2 \, \sqrt{2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt{2} \sinh \left (x\right )^{2} - \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 )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*sqrt(2)*(sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*sqrt(sinh(x)/(cosh(x) - sinh(x))) - (sqrt(2)*cosh(x)^2 + 2*sqr
t(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 - 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)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 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}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((coth(x) + 1)**(3/2), x)

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Giac [B]  time = 1.16432, size = 85, normalized size = 2.58 \begin{align*} -\sqrt{2}{\left (\frac{2}{\sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1} + \log \left ({\left | 2 \, \sqrt{e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\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((1+coth(x))^(3/2),x, algorithm="giac")

[Out]

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