\(\int \sqrt {a \text {csch}^2(x)} \, dx\) [31]

Optimal result

Integrand size = 10, antiderivative size = 26 \[ \int \sqrt {a \text {csch}^2(x)} \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 223, 212} \[ \int \sqrt {a \text {csch}^2(x)} \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right ) \]

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \sqrt {a \text {csch}^2(x)} \, dx=\sqrt {a \text {csch}^2(x)} \left (-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (x) \]

[In]

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

[Out]

Sqrt[a*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. \(2(20)=40\).

Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.58

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.73 \[ \int \sqrt {a \text {csch}^2(x)} \, dx=\left [\sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} - 1\right )} \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}\right ), 2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{x}}{a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )}\right )\right ] \]

[In]

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

[Out]

[sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*(e^(2*x) - 1)*log((cosh(x) + sinh(x) - 1)/(cosh(x) + sinh(x) + 1)), 2*sqrt(
-a)*arctan(sqrt(-a)*sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*(e^(2*x) - 1)*e^x/(a*cosh(x)*e^x + a*e^x*sinh(x)))]

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(a*csch(x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \sqrt {a \text {csch}^2(x)} \, dx=\sqrt {a} \log \left (e^{\left (-x\right )} + 1\right ) - \sqrt {a} \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

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

[Out]

sqrt(a)*log(e^(-x) + 1) - sqrt(a)*log(e^(-x) - 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \sqrt {a \text {csch}^2(x)} \, dx=-\sqrt {a} {\left (\log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int((a/sinh(x)^2)^(1/2), x)