3.24 \(\int \sqrt {-\text {csch}^2(x)} \, dx\)

Optimal. Leaf size=3 \[ \sin ^{-1}(\coth (x)) \]

[Out]

arcsin(coth(x))

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Rubi [A]  time = 0.01, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4122, 216} \[ \sin ^{-1}(\coth (x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[Coth[x]]

Rule 216

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

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Mathematica [B]  time = 0.01, size = 20, normalized size = 6.67 \[ \sinh (x) \sqrt {-\text {csch}^2(x)} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [C]  time = 0.98, size = 15, normalized size = 5.00 \[ -i \, \log \left (e^{x} + 1\right ) + i \, \log \left (e^{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*log(e^x + 1) + I*log(e^x - 1)

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giac [C]  time = 0.14, size = 27, normalized size = 9.00 \[ {\left (i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right )\right )} \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.20, size = 67, normalized size = 22.33 \[ {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-1\right )-{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.56, size = 19, normalized size = 6.33 \[ i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*log(e^(-x) + 1) - I*log(e^(-x) - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.33 \[ \int \sqrt {-\frac {1}{{\mathrm {sinh}\relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-1/sinh(x)^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-csch(x)**2), x)

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