\(\int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx\) [25]

Optimal result

Integrand size = 10, antiderivative size = 13 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, 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, 197} \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]

[In]

Int[1/Sqrt[-Csch[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 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}{\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 = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {-\text {csch}^2(x)}} \]

[In]

Integrate[1/Sqrt[-Csch[x]^2],x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(11)=22\).

Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 4.46

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {1}{2} \, {\left (-i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-x\right )} \]

[In]

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

[Out]

1/2*(-I*e^(2*x) - I)*e^(-x)

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(-csch(x)**2), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=\frac {1}{2} i \, e^{\left (-x\right )} + \frac {1}{2} i \, e^{x} \]

[In]

integrate(1/(-csch(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 = 25, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=-\frac {-i \, e^{\left (-x\right )} - i \, e^{x}}{2 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.38 \[ \int \frac {1}{\sqrt {-\text {csch}^2(x)}} \, dx=-{\mathrm {e}}^{-2\,x}\,\sqrt {-\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}}\,\left (\frac {{\mathrm {e}}^{4\,x}}{4}-\frac {1}{4}\right ) \]

[In]

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

[Out]

-exp(-2*x)*(-1/(exp(-x)/2 - exp(x)/2)^2)^(1/2)*(exp(4*x)/4 - 1/4)