\(\int \frac {1}{\sqrt {a \text {csch}^2(x)}} \, dx\) [32]

Optimal result

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

[Out]

coth(x)/(a*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} \[ \int \frac {1}{\sqrt {a \text {csch}^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {a \text {csch}^2(x)}} \]

[In]

Int[1/Sqrt[a*Csch[x]^2],x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a \text {csch}^2(x)}} \, dx=\frac {\coth (x)}{\sqrt {a \text {csch}^2(x)}} \]

[In]

Integrate[1/Sqrt[a*Csch[x]^2],x]

[Out]

Coth[x]/Sqrt[a*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 = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 4.46

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (11) = 22\).

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 6.38 \[ \int \frac {1}{\sqrt {a \text {csch}^2(x)}} \, dx=\frac {{\left ({\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, {\left (a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(a*csch(x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\sqrt {a \text {csch}^2(x)}} \, dx=-\frac {e^{\left (-x\right )}}{2 \, \sqrt {a}} - \frac {e^{x}}{2 \, \sqrt {a}} \]

[In]

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

[Out]

-1/2*e^(-x)/sqrt(a) - 1/2*e^x/sqrt(a)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (11) = 22\).

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int \frac {1}{\sqrt {a \text {csch}^2(x)}} \, dx=\frac {e^{\left (-x\right )} + e^{x}}{2 \, \sqrt {a} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.54 \[ \int \frac {1}{\sqrt {a \text {csch}^2(x)}} \, dx=-\frac {\left (\frac {{\mathrm {e}}^{-2\,x}}{2}-\frac {{\mathrm {e}}^{2\,x}}{2}\right )\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}}}{2\,\sqrt {a}} \]

[In]

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

[Out]

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