3.31 \(\int \sqrt {a \text {csch}^2(x)} \, dx\)

Optimal. Leaf size=26 \[ -\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4122, 217, 206} \[ -\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

Rule 217

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 4122

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

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.77 \[ \sinh (x) \sqrt {a \text {csch}^2(x)} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a*Csch[x]^2]*Log[Tanh[x/2]]*Sinh[x]

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fricas [B]  time = 1.13, size = 97, normalized size = 3.73 \[ \left [\sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} - 1\right )} \log \left (\frac {\cosh \relax (x) + \sinh \relax (x) - 1}{\cosh \relax (x) + \sinh \relax (x) + 1}\right ), 2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{x}}{a \cosh \relax (x) e^{x} + a e^{x} \sinh \relax (x)}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[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)))]

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giac [A]  time = 0.13, size = 29, normalized size = 1.12 \[ -\sqrt {a} {\left (\log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.23, size = 67, normalized size = 2.58 \[ \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}-1\right )-\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 24, normalized size = 0.92 \[ \sqrt {a} \log \left (e^{\left (-x\right )} + 1\right ) - \sqrt {a} \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {\frac {a}{{\mathrm {sinh}\relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a/sinh(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \operatorname {csch}^{2}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*csch(x)**2), x)

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