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

Optimal. Leaf size=31 \[ \frac {1}{2} \tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\text {csch}^2(x)}}\right )-\frac {1}{2} \coth (x) \sqrt {\text {csch}^2(x)} \]

[Out]

1/2*arctanh(coth(x)/(csch(x)^2)^(1/2))-1/2*coth(x)*(csch(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3657, 4122, 195, 217, 206} \[ \frac {1}{2} \tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\text {csch}^2(x)}}\right )-\frac {1}{2} \coth (x) \sqrt {\text {csch}^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Coth[x]^2)^(3/2),x]

[Out]

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

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 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 217

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

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
]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 40, normalized size = 1.29 \[ -\frac {1}{8} \sinh (x) \sqrt {\text {csch}^2(x)} \left (\text {csch}^2\left (\frac {x}{2}\right )+\text {sech}^2\left (\frac {x}{2}\right )+4 \log \left (\tanh \left (\frac {x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Coth[x]^2)^(3/2),x]

[Out]

-1/8*(Sqrt[Csch[x]^2]*(Csch[x/2]^2 + 4*Log[Tanh[x/2]] + Sech[x/2]^2)*Sinh[x])

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 211, normalized size = 6.81 \[ -\frac {2 \, \cosh \relax (x)^{3} + 6 \, \cosh \relax (x) \sinh \relax (x)^{2} + 2 \, \sinh \relax (x)^{3} - {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + 2 \, \cosh \relax (x)}{2 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*cosh(x)^3 + 6*cosh(x)*sinh(x)^2 + 2*sinh(x)^3 - (cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*co
sh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) + 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)*log(cosh(x) + sinh(x) - 1) + 2*(3*cosh(x)^2 + 1)*sinh(x) + 2*cosh(x))/(cosh(x)^4 + 4*cosh(x)*si
nh(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 = 0.12, size = 52, normalized size = 1.68 \[ -\frac {1}{4} \, {\left (\frac {4 \, {\left (e^{\left (-x\right )} + e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]

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

________________________________________________________________________________________

maple [A]  time = 0.13, size = 28, normalized size = 0.90 \[ -\frac {\coth \relax (x ) \sqrt {-1+\coth ^{2}\relax (x )}}{2}+\frac {\ln \left (\coth \relax (x )+\sqrt {-1+\coth ^{2}\relax (x )}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 46, normalized size = 1.48 \[ -\frac {e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.20, size = 27, normalized size = 0.87 \[ \frac {\ln \left (\mathrm {coth}\relax (x)+\sqrt {{\mathrm {coth}\relax (x)}^2-1}\right )}{2}-\frac {\mathrm {coth}\relax (x)\,\sqrt {{\mathrm {coth}\relax (x)}^2-1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((coth(x)^2 - 1)^(3/2),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\coth ^{2}{\relax (x )} - 1\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((coth(x)**2 - 1)**(3/2), x)

________________________________________________________________________________________