3.12 \(\int (1-\coth ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{2} \sin ^{-1}(\coth (x))+\frac{1}{2} \coth (x) \sqrt{-\text{csch}^2(x)} \]

[Out]

ArcSin[Coth[x]]/2 + (Coth[x]*Sqrt[-Csch[x]^2])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0216834, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3657, 4122, 195, 216} \[ \frac{1}{2} \sin ^{-1}(\coth (x))+\frac{1}{2} \coth (x) \sqrt{-\text{csch}^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Coth[x]^2)^(3/2),x]

[Out]

ArcSin[Coth[x]]/2 + (Coth[x]*Sqrt[-Csch[x]^2])/2

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0589139, size = 41, normalized size = 1.71 \[ \frac{1}{4} \text{csch}\left (\frac{x}{2}\right ) \sqrt{-\text{csch}^2(x)} \text{sech}\left (\frac{x}{2}\right ) \left (\cosh (x)+\sinh ^2(x) \log \left (\tanh \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Coth[x]^2)^(3/2),x]

[Out]

(Csch[x/2]*Sqrt[-Csch[x]^2]*Sech[x/2]*(Cosh[x] + Log[Tanh[x/2]]*Sinh[x]^2))/4

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 21, normalized size = 0.9 \begin{align*}{\frac{{\rm coth} \left (x\right )}{2}\sqrt{1- \left ({\rm coth} \left (x\right ) \right ) ^{2}}}+{\frac{\arcsin \left ({\rm coth} \left (x\right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-coth(x)^2)^(3/2),x)

[Out]

1/2*(1-coth(x)^2)^(1/2)*coth(x)+1/2*arcsin(coth(x))

________________________________________________________________________________________

Maxima [C]  time = 1.90969, size = 66, normalized size = 2.75 \begin{align*} \frac{i \, e^{\left (-x\right )} + i \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac{1}{2} i \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac{1}{2} i \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*e^(-x) + I*e^(-3*x))/(2*e^(-2*x) - e^(-4*x) - 1) + 1/2*I*log(e^(-x) + 1) - 1/2*I*log(e^(-x) - 1)

________________________________________________________________________________________

Fricas [A]  time = 3.04079, size = 4, normalized size = 0.17 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((1 - coth(x)**2)**(3/2), x)

________________________________________________________________________________________

Giac [C]  time = 1.14349, size = 81, normalized size = 3.38 \begin{align*} -\frac{1}{4} \,{\left (\frac{4 \,{\left (i \, e^{\left (-x\right )} + i \, e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - i \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + i \, \log \left (e^{\left (-x\right )} + e^{x} - 2\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)^2)^(3/2),x, algorithm="giac")

[Out]

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