\(\int \sqrt {-\text {csch}^2(x)} \, dx\) [24]

Optimal result

Integrand size = 10, antiderivative size = 3 \[ \int \sqrt {-\text {csch}^2(x)} \, dx=\arcsin (\coth (x)) \]

[Out]

arcsin(coth(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4207, 222} \[ \int \sqrt {-\text {csch}^2(x)} \, dx=\arcsin (\coth (x)) \]

[In]

Int[Sqrt[-Csch[x]^2],x]

[Out]

ArcSin[Coth[x]]

Rule 222

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]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(30\) vs. \(2(3)=6\).

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 10.00 \[ \int \sqrt {-\text {csch}^2(x)} \, dx=\sqrt {-\text {csch}^2(x)} \left (-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (x) \]

[In]

Integrate[Sqrt[-Csch[x]^2],x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(3)=6\).

Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 22.33

method	result	size
risch	\(\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}-1\right )-\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}+1\right )\)	\(67\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 5.00 \[ \int \sqrt {-\text {csch}^2(x)} \, dx=-i \, \log \left (e^{x} + 1\right ) + i \, \log \left (e^{x} - 1\right ) \]

[In]

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

[Out]

-I*log(e^x + 1) + I*log(e^x - 1)

Sympy [F]

\[ \int \sqrt {-\text {csch}^2(x)} \, dx=\int \sqrt {- \operatorname {csch}^{2}{\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(-csch(x)**2), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 6.33 \[ \int \sqrt {-\text {csch}^2(x)} \, dx=i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

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

[Out]

I*log(e^(-x) + 1) - I*log(e^(-x) - 1)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 9.00 \[ \int \sqrt {-\text {csch}^2(x)} \, dx={\left (i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \]

[In]

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

[Out]

(I*log(e^x + 1) - I*log(abs(e^x - 1)))*sgn(-e^(3*x) + e^x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {-\text {csch}^2(x)} \, dx=\int \sqrt {-\frac {1}{{\mathrm {sinh}\left (x\right )}^2}} \,d x \]

[In]

int((-1/sinh(x)^2)^(1/2),x)

[Out]

int((-1/sinh(x)^2)^(1/2), x)