3.23 \(\int \sqrt {-1+\text {csch}^2(x)} \, dx\)

Optimal. Leaf size=33 \[ -\tan ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )-\tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right ) \]

[Out]

-arctan(coth(x)/(-2+coth(x)^2)^(1/2))-arctanh(coth(x)/(-2+coth(x)^2)^(1/2))

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4128, 402, 217, 206, 377, 203} \[ -\tan ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )-\tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Csch[x]^2],x]

[Out]

-ArcTan[Coth[x]/Sqrt[-2 + Coth[x]^2]] - ArcTanh[Coth[x]/Sqrt[-2 + Coth[x]^2]]

Rule 203

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

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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 4128

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [B]  time = 0.04, size = 68, normalized size = 2.06 \[ \frac {\sqrt {2} \sinh (x) \sqrt {\text {csch}^2(x)-1} \left (\log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)-3}\right )+\tan ^{-1}\left (\frac {\sqrt {2} \cosh (x)}{\sqrt {\cosh (2 x)-3}}\right )\right )}{\sqrt {\cosh (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + Csch[x]^2],x]

[Out]

(Sqrt[2]*Sqrt[-1 + Csch[x]^2]*(ArcTan[(Sqrt[2]*Cosh[x])/Sqrt[-3 + Cosh[2*x]]] + Log[Sqrt[2]*Cosh[x] + Sqrt[-3
+ Cosh[2*x]]])*Sinh[x])/Sqrt[-3 + Cosh[2*x]]

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fricas [B]  time = 0.42, size = 358, normalized size = 10.85 \[ \frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \sqrt {-\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + 2 \, \cosh \relax (x)\right )} \sinh \relax (x) - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {2} {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \sqrt {-\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 6 \, {\left (\cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 6 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x) + 1}\right ) - \frac {1}{2} \, \log \left (\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + \sqrt {2} \sqrt {-\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) + \frac {1}{2} \, \log \left (\frac {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - \sqrt {2} \sqrt {-\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} + 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^
2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 2)*sinh(x)
^2 + 4*cosh(x)^2 + 4*(cosh(x)^3 + 2*cosh(x))*sinh(x) - 1)) + 1/2*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x)
+ sinh(x)^2 - 1)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(cosh(x)^4 +
4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 6*(cosh(x)^2 - 1)*sinh(x)^2 - 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x))*sinh(x
) + 1)) - 1/2*log((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + sqrt(2)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh
(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 1/2*log((cosh(x)^2
+ 2*cosh(x)*sinh(x) + sinh(x)^2 - sqrt(2)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) +
sinh(x)^2)) + 1)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2))

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giac [B]  time = 0.19, size = 157, normalized size = 4.76 \[ \frac {1}{2} \, {\left (\arcsin \left (\frac {1}{4} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 3\right )}\right ) + 2 \, \arctan \left (-2 \, \sqrt {2} - \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3}\right ) - 2 \, \log \left ({\left | -\sqrt {2} - \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3} + 1 \right |}\right ) + 2 \, \log \left ({\left | -\sqrt {2} - \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3} - 1 \right |}\right )\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(arcsin(1/4*sqrt(2)*(e^(2*x) - 3)) + 2*arctan(-2*sqrt(2) - 3*(2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/
(e^(2*x) - 3)) - 2*log(abs(-sqrt(2) - (2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/(e^(2*x) - 3) + 1)) + 2*log
(abs(-sqrt(2) - (2*sqrt(2) - sqrt(-e^(4*x) + 6*e^(2*x) - 1))/(e^(2*x) - 3) - 1)))*sgn(-e^(2*x) + 1)

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maple [F]  time = 0.33, size = 0, normalized size = 0.00 \[ \int \sqrt {-1+\mathrm {csch}\relax (x )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {csch}\relax (x)^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(csch(x)^2 - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {\frac {1}{{\mathrm {sinh}\relax (x)}^2}-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sinh(x)^2 - 1)^(1/2),x)

[Out]

int((1/sinh(x)^2 - 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {csch}^{2}{\relax (x )} - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(csch(x)**2 - 1), x)

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