\(\int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx\) [14]

Optimal result

Integrand size = 12, antiderivative size = 13 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3738, 4207, 197} \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]

[In]

Int[1/Sqrt[1 - Coth[x]^2],x]

[Out]

Coth[x]/Sqrt[-Csch[x]^2]

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 3738

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 4207

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]

[In]

Integrate[1/Sqrt[1 - Coth[x]^2],x]

[Out]

Coth[x]/Sqrt[-Csch[x]^2]

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {\coth \left (x \right )}{\sqrt {1-\coth \left (x \right )^{2}}}\)	\(14\)
default	\(\frac {\coth \left (x \right )}{\sqrt {1-\coth \left (x \right )^{2}}}\)	\(14\)
risch	\(\frac {{\mathrm e}^{2 x}}{2 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {1}{2 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}\)	\(58\)

[In]

int(1/(1-coth(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (11) = 22\).

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.08 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=-{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sqrt {-\frac {e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{\left (-x\right )} \]

[In]

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

[Out]

-(cosh(x)*e^(2*x) - cosh(x))*sqrt(-e^(2*x)/(e^(4*x) - 2*e^(2*x) + 1))*e^(-x)

Sympy [F]

\[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\int \frac {1}{\sqrt {1 - \coth ^{2}{\left (x \right )}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(1 - coth(x)**2), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=\frac {1}{2} i \, e^{\left (-x\right )} + \frac {1}{2} i \, e^{x} \]

[In]

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

[Out]

1/2*I*e^(-x) + 1/2*I*e^x

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=-\frac {-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt {1-\coth ^2(x)}} \, dx=-\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {-\frac {1}{{\mathrm {cosh}\left (x\right )}^2-1}} \]

[In]

int(1/(1 - coth(x)^2)^(1/2),x)

[Out]

-cosh(x)*sinh(x)*(-1/(cosh(x)^2 - 1))^(1/2)